Content check — Algebra and Trig, Chapter 9: Trigonometric Identities and Equations

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 9: Trigonometric Identities and Equations

234 problems · 7 figures

Try It #1 §9.1 source ↗

Verify the identity

\csc θ \cos θ \tan θ =

1.

Answer

\begin{matrix} \csc θ \cos θ \tan θ & = & (\frac{1}{\sin θ}) \cos θ (\frac{\sin θ}{\cos θ}) \\ = & \frac{\cos θ}{\sin θ} (\frac{\sin θ}{\cos θ}) \\ = & \frac{\sin θ \cos θ}{\sin θ \cos θ} \\ = & 1 \end{matrix}

Try It #2 §9.1 source ↗

Show that

\frac{\cot θ}{\csc θ} =

\cos θ .

Answer

\begin{matrix} \frac{\cot θ}{\csc θ} & = & \frac{\frac{\cos θ}{\sin θ}}{\frac{1}{\sin θ}} \\ = & \frac{\cos θ}{\sin θ} \cdot \frac{\sin θ}{1} \\ = & \cos θ \end{matrix}

Try It #3 §9.1 source ↗

Verify the identity

\frac{\sin^{2} θ - 1}{\tan θ \sin θ - \tan θ} =

\frac{\sin θ + 1}{\tan θ} .

Answer

\begin{matrix} \frac{\sin^{2} θ - 1}{\tan θ \sin θ - \tan θ} & = & \frac{(\sin θ + 1) (\sin θ - 1)}{\tan θ (\sin θ - 1)} \\ = & \frac{\sin θ + 1}{\tan θ} \end{matrix}

Try It #4 §9.1 source ↗

Rewrite the trigonometric expression using the difference of squares:

25 - 9 \sin^{2} θ .

Answer

This is a difference of squares formula:

25 -

9 \sin^{2} θ =

(5 - 3 \sin θ) (5 + 3 \sin θ) .

Try It #5 §9.1 source ↗

Use algebraic techniques to verify the identity: $\frac{\cos θ}{1 + \sin θ} =$ $\frac{1 - \sin θ}{\cos θ} .$

(Hint: Multiply the numerator and denominator on the left side by $1 - \sin θ .)$

Answer

\begin{matrix} \frac{\cos θ}{1 + \sin θ} (\frac{1 - \sin θ}{1 - \sin θ}) & = & \frac{\cos θ (1 - \sin θ)}{1 - \sin^{2} θ} \\ = & \frac{\cos θ (1 - \sin θ)}{\cos^{2} θ} \\ = & \frac{1 - \sin θ}{\cos θ} \end{matrix}

9.1-1 §9.1 source ↗

We know

g (x) = \cos x

is an even function, and

f (x) = \sin x

and

h (x) = \tan x

are odd functions. What about

G (x) =

\cos^{2} x,

F (x) =

\sin^{2} x,

and

H (x) = \tan^{2} x ?

Are they even, odd, or neither? Why?

Answer

All three functions, $F$ , $G$ , and $H$ , are even.

This is because $F (- x) =$ $\sin (- x) \sin (- x) =$ $(- \sin x) (- \sin x) =$ $\sin^{2} x =$ $F (x)$ , $G (- x) =$ $\cos (- x) \cos (- x) =$ $\cos x \cos x =$ $\cos^{2} x =$ $G (x)$ and $H (- x) =$ $\tan (- x) \tan (- x) =$ $(- \tan x) (- \tan x) =$ $\tan^{2} x =$ $H (x) .$

9.1-3 §9.1 source ↗

After examining the reciprocal identity for

\sec t,

explain why the function is undefined at certain points.

Answer

When

\cos t = 0,

then

\sec t = \frac{1}{0},

which is undefined.

9.1-5 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$\sin x \cos x \sec x$

Answer

\sin x

9.1-7 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$\tan x \sin x +$ $\sec x \cos^{2} x$

Answer

\sec x

9.1-9 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{\cot t + \tan t}{\sec (- t)}$

Answer

\csc t

9.1-11 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$- \tan (- x) \cot (- x)$

Answer

−1

9.1-13 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{1 + \tan^{2} θ}{\csc^{2} θ} +$ $\sin^{2} θ +$ $\frac{1}{\sec^{2} θ}$

Answer

\sec^{2} x

9.1-15 §9.1 source ↗

For the following exercises, use the fundamental identities to fully simplify the expression.

$\frac{1 - \cos^{2} x}{\tan^{2} x} +$ $2 \sin^{2} x$

Answer

\sin^{2} x + 1

9.1-17 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{\sec x + \csc x}{1 + \tan x};$ $\sin x$

Answer

\frac{1}{\sin x}

9.1-19 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{1}{\sin x \cos x} -$ $\cot x;$ $\cot x$

Answer

\frac{1}{\cot x}

9.1-21 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$(\sec x + \csc x) (\sin x + \cos x) -$ $2 -$ $\cot x;$ $\tan x$

Answer

\tan x

9.1-23 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{1 - \sin x}{1 + \sin x} -$ $\frac{1 + \sin x}{1 - \sin x};$ $\sec x and \tan x$

Answer

- 4 \sec x \tan x

9.1-25 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\sec x; \cot x$

Answer

\pm \sqrt{\frac{1}{\cot^{2} x} + 1}

9.1-27 §9.1 source ↗

For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\cot x; \sin x$

Answer

\frac{\pm \sqrt{1 - \sin^{2} x}}{\sin x}

9.1-29 §9.1 source ↗

For the following exercises, verify the identity.

$\cos x -$ $\cos^{3} x =$ $\cos x \sin^{2} x$

Answer

Answers will vary. Sample proof:

$\begin{matrix} \cos x - \cos^{3} x & = & \cos x (1 - \cos^{2} x) \\ = & \cos x \sin^{2} x \end{matrix}$

9.1-31 §9.1 source ↗

For the following exercises, verify the identity.

$\frac{1 + \sin^{2} x}{\cos^{2} x} =$ $\frac{1}{\cos^{2} x} +$ $\frac{\sin^{2} x}{\cos^{2} x} =$ $1 +$ $2 \tan^{2} x$

Answer

Answers will vary. Sample proof:

\frac{1 + \sin^{2} x}{\cos^{2} x} =

\frac{1}{\cos^{2} x} +

\frac{\sin^{2} x}{\cos^{2} x} =

\sec^{2} x +

\tan^{2} x =

\tan^{2} x +

1 +

\tan^{2} x =

1 +

2 \tan^{2} x

9.1-33 §9.1 source ↗

For the following exercises, verify the identity.

$\cos^{2} x -$ $\tan^{2} x =$ $2 -$ $\sin^{2} x -$ $\sec^{2} x$

Answer

Answers will vary. Sample proof:

\cos^{2} x -

\tan^{2} x =

1 -

\sin^{2} x -

(\sec^{2} x - 1) =

1 -

\sin^{2} x -

\sec^{2} x +

1 =

2 -

\sin^{2} x -

\sec^{2} x

9.1-35 §9.1 source ↗

For the following exercises, prove or disprove the identity.

$\csc^{2} x (1 + \sin^{2} x) =$ $\cot^{2} x$

Answer

False

9.1-37 §9.1 source ↗

For the following exercises, prove or disprove the identity.

$\frac{\tan x}{\sec x} \sin (- x) =$ $\cos^{2} x$

Answer

False

9.1-39 §9.1 source ↗

For the following exercises, prove or disprove the identity.

$\frac{1 + \sin x}{\cos x} =$ $\frac{\cos x}{1 + \sin (- x)}$

Answer

Proved with negative and Pythagorean identities

9.1-41 §9.1 source ↗

For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

$3 \sin^{2} θ +$ $4 \cos^{2} θ =$ $3 +$ $\cos^{2} θ$

Answer

True

3 \sin^{2} θ +

4 \cos^{2} θ =

3 \sin^{2} θ +

3 \cos^{2} θ +

\cos^{2} θ =

3 (\sin^{2} θ + \cos^{2} θ) +

\cos^{2} θ =

3 +

\cos^{2} θ

Try It #1 §9.2 source ↗

Find the exact value of

\cos (\frac{π}{3} - \frac{π}{4}) .

Answer

\frac{\sqrt{2} + \sqrt{6}}{4}

Try It #2 §9.2 source ↗

Find the exact value of

\cos (105°) .

Answer

\frac{\sqrt{2} - \sqrt{6}}{4}

Try It #3 §9.2 source ↗

Find the exact value of

\tan (\frac{2 π}{3} + \frac{π}{4}) .

Answer

\frac{1 - \sqrt{3}}{1 + \sqrt{3}}

Try It #4 §9.2 source ↗

Write

\sin \frac{π}{7}

in terms of its cofunction.

Answer

\cos (\frac{5 π}{14})

Try It #5 §9.2 source ↗

Verify the identity:

\tan (π - θ) =

-

\tan θ .

Answer

\begin{matrix} \tan (π - θ) & = & \frac{\tan (π) - \tan θ}{1 + \tan (π) \tan θ} \\ = & \frac{0 - \tan θ}{1 + 0 \cdot \tan θ} \\ = & - \tan θ \end{matrix}

9.2-1 §9.2 source ↗

Explain the basis for the cofunction identities and when they apply.

Answer

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures

x,

the second angle measures

\frac{π}{2} - x .

Then

\sin x =

\cos (\frac{π}{2} - x) .

The same holds for the other cofunction identities. The key is that the angles are complementary.

9.2-3 §9.2 source ↗

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for

f (x) = \sin (x)

and

g (x) =

\cos (x) .

(Hint:

0 - x = - x

)

Answer

\sin (- x) =

-

\sin x,

\sin x

is odd.

\cos (- x) =

\cos (0 - x) =

\cos x,

\cos x

is even.

9.2-5 §9.2 source ↗

For the following exercises, find the exact value.

$\cos (\frac{π}{12})$

Answer

\frac{\sqrt{2} + \sqrt{6}}{4}

9.2-7 §9.2 source ↗

For the following exercises, find the exact value.

$\sin (\frac{11 π}{12})$

Answer

\frac{\sqrt{6} - \sqrt{2}}{4}

9.2-9 §9.2 source ↗

For the following exercises, find the exact value.

$\tan (\frac{19 π}{12})$

Answer

- 2 - \sqrt{3}

9.2-11 §9.2 source ↗

For the following exercises, rewrite in terms of $\sin x$ and $\cos x .$

$\sin (x - \frac{3 π}{4})$

Answer

-

\frac{\sqrt{2}}{2} \sin x -

\frac{\sqrt{2}}{2} \cos x

9.2-13 §9.2 source ↗

For the following exercises, rewrite in terms of $\sin x$ and $\cos x .$

$\cos (x + \frac{2 π}{3})$

Answer

-

\frac{1}{2} \cos x -

\frac{\sqrt{3}}{2} \sin x

9.2-15 §9.2 source ↗

For the following exercises, simplify the given expression.

$\sec (\frac{π}{2} - θ)$

Answer

\csc θ

9.2-17 §9.2 source ↗

For the following exercises, simplify the given expression.

$\tan (\frac{π}{2} - x)$

Answer

\cot x

9.2-19 §9.2 source ↗

For the following exercises, simplify the given expression.

$\frac{\tan (\frac{3}{2} x) - \tan (\frac{7}{5} x)}{1 + \tan (\frac{3}{2} x) \tan (\frac{7}{5} x)}$

Answer

\tan (\frac{x}{10})

9.2-21 §9.2 source ↗

For the following exercises, find the requested information.

Given that $\sin a = \frac{4}{5},$ and $\cos b = \frac{1}{3},$ with $a$ and $b$ both in the interval $[0, \frac{π}{2}),$ find $\sin (a - b)$ and $\cos (a + b) .$

Answer

\begin{matrix} \sin (a - b) & = & (\frac{4}{5}) (\frac{1}{3}) - (\frac{3}{5}) (\frac{2 \sqrt{2}}{3}) & = & \frac{4 - 6 \sqrt{2}}{15} \\ \cos (a + b) & = & (\frac{3}{5}) (\frac{1}{3}) - (\frac{4}{5}) (\frac{2 \sqrt{2}}{3}) & = & \frac{3 - 8 \sqrt{2}}{15} \end{matrix}

9.2-23 §9.2 source ↗

For the following exercises, find the exact value of each expression.

$\cos (\cos^{- 1} (\frac{\sqrt{2}}{2}) + \sin^{- 1} (\frac{\sqrt{3}}{2}))$

Answer

\frac{\sqrt{2} - \sqrt{6}}{4}

9.2-25 §9.2 source ↗

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\cos (\frac{π}{2} - x)$

Answer

$\sin x$

9.2-27 §9.2 source ↗

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\tan (\frac{π}{3} + x)$

Answer

$\cot (\frac{π}{6} - x)$

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.

9.2-29 §9.2 source ↗

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\tan (\frac{π}{4} - x)$

Answer

$\cot (\frac{π}{4} + x)$

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.

9.2-31 §9.2 source ↗

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

$\sin (\frac{π}{4} + x)$

Answer

$\frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}}$

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.

9.2-33 §9.2 source ↗

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think $2 x = x + x .$ )

$f (x) =$ $\sin (4 x) -$ $\sin (3 x) \cos x,$ $g (x) =$ $\sin x \cos (3 x)$

Answer

They are the same.

9.2-35 §9.2 source ↗

$f (x) =$ $\sin (3 x) \cos (6 x),$ $g (x) =$ $-$ $\sin (3 x) \cos (6 x)$

Answer

They are the different, try

g (x) =

\sin (9 x) -

\cos (3 x) \sin (6 x) .

9.2-37 §9.2 source ↗

$f (x) =$ $\sin (2 x),$ $g (x) =$ $2 \sin x \cos x$

Answer

They are the same.

9.2-39 §9.2 source ↗

$f (θ) =$ $\tan (2 θ),$ $g (θ) =$ $\frac{\tan θ}{1 + \tan^{2} θ}$

Answer

They are the different, try

g (θ) =

\frac{2 \tan θ}{1 - \tan^{2} θ} .

9.2-41 §9.2 source ↗

$f (x) =$ $\tan (- x),$ $g (x) =$ $\frac{\tan x - \tan (2 x)}{1 - \tan x \tan (2 x)}$

Answer

They are different, try

g (x) =

\frac{\tan x - \tan (2 x)}{1 + \tan x \tan (2 x)} .

9.2-43 §9.2 source ↗

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\sin (195°)$

Answer

-

\frac{\sqrt{3} - 1}{2 \sqrt{2}},

or -

0.2588

9.2-45 §9.2 source ↗

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

$\cos (345°)$

Answer

\frac{1 + \sqrt{3}}{2 \sqrt{2}},

or 0.9659

9.2-47 §9.2 source ↗

For the following exercises, prove the identities provided.

$\tan (x + \frac{π}{4}) =$ $\frac{\tan x + 1}{1 - \tan x}$

Answer

\begin{matrix} \tan (x + \frac{π}{4}) & = \\ \frac{\tan x + \tan (\frac{π}{4})}{1 - \tan x \tan (\frac{π}{4})} & = \\ \frac{\tan x + 1}{1 - \tan x (1)} & = & \frac{\tan x + 1}{1 - \tan x} \end{matrix}

9.2-49 §9.2 source ↗

For the following exercises, prove the identities provided.

$\frac{\cos (a + b)}{\cos a \cos b} =$ $1 -$ $\tan a \tan b$

Answer

\begin{matrix} \frac{\cos (a + b)}{\cos a \cos b} & = \\ \frac{\cos a \cos b}{\cos a \cos b} - \frac{\sin a \sin b}{\cos a \cos b} & = & 1 - \tan a \tan b \end{matrix}

9.2-51 §9.2 source ↗

For the following exercises, prove the identities provided.

$\frac{\cos (x + h) - \cos x}{h} =$ $\cos x \frac{\cos h - 1}{h} -$ $\sin x \frac{\sin h}{h}$

Answer

\begin{matrix} \frac{\cos (x + h) - \cos x}{h} & = \\ \frac{\cos x \cosh - \sin x \sinh - \cos x}{h} & = \\ \frac{\cos x (\cosh - 1) - \sin x \sinh}{h} & = & \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h} \end{matrix}

9.2-53 §9.2 source ↗

For the following exercises, prove or disprove the statements.

$\tan (u - v) =$ $\frac{\tan u - \tan v}{1 + \tan u \tan v}$

Answer

True

9.2-55 §9.2 source ↗

For the following exercises, prove or disprove the statements.

If $α, β,$ and $γ$ are angles in the same triangle, then prove or disprove $\sin (α + β) =$ $\sin γ .$

Answer

True. Note that

\sin (α + β) =

\sin (π - γ)

and expand the right hand side.

Try It #1 §9.3 source ↗

Given

\sin α = \frac{5}{8},

with

α

in quadrant I, find

\cos (2 α) .

Answer

\cos (2 α) = \frac{7}{32}

Try It #2 §9.3 source ↗

Verify the identity:

\cos^{4} θ -

\sin^{4} θ =

\cos (2 θ) .

Answer

\cos^{4} θ -

\sin^{4} θ =

(\cos^{2} θ + \sin^{2} θ) (\cos^{2} θ - \sin^{2} θ) =

\cos (2 θ)

Try It #3 §9.3 source ↗

Verify the identity:

\cos (2 θ) \cos θ =

\cos^{3} θ -

\cos θ \sin^{2} θ .

Answer

\cos (2 θ) \cos θ =

(\cos^{2} θ - \sin^{2} θ) \cos θ =

\cos^{3} θ -

\cos θ \sin^{2} θ

Try It #4 §9.3 source ↗

Use the power-reducing formulas to prove that

10 \cos^{4} x =

\frac{15}{4} +

5 \cos (2 x) +

\frac{5}{4} \cos (4 x) .

Answer

\begin{matrix} 10 \cos^{4} x & = & 10 {(\cos^{2} x)}^{2} \\ = & 10 {[\frac{1 + \cos (2 x)}{2}]}^{2} & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{10}{4} [1 + 2 \cos (2 x) + \cos^{2} (2 x)] \\ = & \frac{10}{4} + \frac{10}{2} \cos (2 x) + \frac{10}{4} (\frac{1 + \cos 2 (2 x)}{2}) & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{10}{4} + \frac{10}{2} \cos (2 x) + \frac{10}{8} + \frac{10}{8} \cos (4 x) \\ = & \frac{30}{8} + 5 \cos (2 x) + \frac{10}{8} \cos (4 x) \\ = & \frac{15}{4} + 5 \cos (2 x) + \frac{5}{4} \cos (4 x) \end{matrix}

Try It #5 §9.3 source ↗

Given that

\sin α = - \frac{4}{5}

and

α

lies in quadrant IV, find the exact value of

\cos (\frac{α}{2}) .

Answer

- \frac{2}{\sqrt{5}}

9.3-1 §9.3 source ↗

Explain how to determine the reduction identities from the double-angle identity

\cos (2 x) =

\cos^{2} x -

\sin^{2} x .

Answer

Use the Pythagorean identities and isolate the squared term.

9.3-3 §9.3 source ↗

We can determine the half-angle formula for

\tan (\frac{x}{2}) =

\frac{\sqrt{1 - \cos x}}{\sqrt{1 + \cos x}}

by dividing the formula for

\sin (\frac{x}{2})

\cos (\frac{x}{2}) .

Explain how to determine two formulas for

\tan (\frac{x}{2})

that do not involve any square roots.

Answer

\frac{1 - \cos x}{\sin x},

\frac{\sin x}{1 + \cos x},

multiplying the top and bottom by

\sqrt{1 - \cos x}

and

\sqrt{1 + \cos x},

respectively.

9.3-5 §9.3 source ↗

For the following exercises, find the exact values of a) $\sin (2 x),$ b) $\cos (2 x),$ and c) $\tan (2 x)$ without solving for $x .$

If $\sin x = \frac{1}{8},$ and $x$ is in quadrant I.

Answer

\frac{3 \sqrt{7}}{32}

\frac{31}{32}

\frac{3 \sqrt{7}}{31}

9.3-7 §9.3 source ↗

For the following exercises, find the exact values of a) $\sin (2 x),$ b) $\cos (2 x),$ and c) $\tan (2 x)$ without solving for $x .$

If $\cos x = - \frac{1}{2},$ and $x$ is in quadrant III.

Answer

\frac{\sqrt{3}}{2}

- \frac{1}{2}

- \sqrt{3}

9.3-9 §9.3 source ↗

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

$\cos (2 θ) = \frac{3}{5}$ and $90° \leq θ \leq 180°$

Answer

\cos θ =

-

\frac{2 \sqrt{5}}{5},

\sin θ =

\frac{\sqrt{5}}{5},

\tan θ =

-

\frac{1}{2},

\csc θ =

\sqrt{5},

\sec θ =

-

\frac{\sqrt{5}}{2},

\cot θ =

-

2

9.3-11 §9.3 source ↗

For the following exercises, simplify to one trigonometric expression.

$2 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Answer

\sin (\frac{π}{2})

9.3-13 §9.3 source ↗

For the following exercises, find the exact value using half-angle formulas.

$\sin (\frac{π}{8})$

Answer

\frac{\sqrt{2 - \sqrt{2}}}{2}

9.3-15 §9.3 source ↗

For the following exercises, find the exact value using half-angle formulas.

$\sin (\frac{11 π}{12})$

Answer

\frac{\sqrt{2 - \sqrt{3}}}{2}

9.3-17 §9.3 source ↗

For the following exercises, find the exact value using half-angle formulas.

$\tan (\frac{5 π}{12})$

Answer

2 + \sqrt{3}

9.3-19 §9.3 source ↗

For the following exercises, find the exact value using half-angle formulas.

$\tan (- \frac{3 π}{8})$

Answer

- 1 - \sqrt{2}

9.3-21 §9.3 source ↗

For the following exercises, find the exact values of a) $\sin (\frac{x}{2}),$ b) $\cos (\frac{x}{2}),$ and c) $\tan (\frac{x}{2})$ without solving for $x,$ when $0 \leq x < 2 π$

If $\sin x = - \frac{12}{13},$ and $x$ is in quadrant III.

Answer

\frac{3 \sqrt{13}}{13}

- \frac{2 \sqrt{13}}{13}

- \frac{3}{2}

9.3-23 §9.3 source ↗

For the following exercises, find the exact values of a) $\sin (\frac{x}{2}),$ b) $\cos (\frac{x}{2}),$ and c) $\tan (\frac{x}{2})$ without solving for $x,$ when $0 \leq x < 2 π$

If $\sec x = - 4,$ and $x$ is in quadrant II.

Answer

\frac{\sqrt{10}}{4}

\frac{\sqrt{6}}{4}

\frac{\sqrt{15}}{3}

9.3-25 §9.3 source ↗

For the following exercises, use Figure 5 to find the requested half and double angles.

Image of a right triangle. The base is length 12, and the height is length 5. The angle between the base and the height is 90 degrees, the angle between the base and the hypotenuse is theta, and the angle between the height and the hypotenuse is alpha degrees.

Figure 5

Find $\sin (2 α),$ $\cos (2 α),$ and $\tan (2 α).$

Answer

\frac{120}{169},

- \frac{119}{169},

- \frac{120}{119}

9.3-27 §9.3 source ↗

For the following exercises, use Figure 5 to find the requested half and double angles.

Figure 5

Find $\sin (\frac{α}{2}),$ $\cos (\frac{α}{2}),$ and $\tan (\frac{α}{2}) .$

Answer

\frac{2 \sqrt{13}}{13},

\frac{3 \sqrt{13}}{13},

\frac{2}{3}

9.3-29 §9.3 source ↗

For the following exercises, simplify each expression. Do not evaluate.

$2 \cos^{2} (37°) -$ $1$

Answer

\cos (74°)

9.3-31 §9.3 source ↗

For the following exercises, simplify each expression. Do not evaluate.

$\cos^{2} (9 x) -$ $\sin^{2} (9 x)$

Answer

\cos (18 x)

9.3-33 §9.3 source ↗

For the following exercises, simplify each expression. Do not evaluate.

$6 \sin (5 x) \cos (5 x)$

Answer

3 \sin (10 x)

9.3-35 §9.3 source ↗

For the following exercises, prove the given identity.

$\sin (2 x) =$ $-$ $2 \sin (- x) \cos (- x)$

Answer

-

2 \sin (- x) \cos (- x) =

-

2 (- \sin (x) \cos (x)) =

\sin (2 x)

9.3-37 §9.3 source ↗

For the following exercises, prove the given identity.

$\frac{\sin (2 θ)}{1 + \cos (2 θ)} \tan^{2} θ =$ $\tan^{3} θ$

Answer

\begin{matrix} \frac{\sin (2 θ)}{1 + \cos (2 θ)} \tan^{2} θ & = & \frac{2 \sin (θ) \cos (θ)}{1 + \cos^{2} θ - \sin^{2} θ} \tan^{2} θ = \\ \frac{2 \sin (θ) \cos (θ)}{2 \cos^{2} θ} \tan^{2} θ & = & \frac{\sin (θ)}{\cos θ} \tan^{2} θ = \\ \cot (θ) \tan^{2} θ & = & \tan^{3} θ \end{matrix}

9.3-39 §9.3 source ↗

For the following exercises, rewrite the expression with an exponent no higher than 1.

$\cos^{2} (6 x)$

Answer

\frac{1 + \cos (12 x)}{2}

9.3-41 §9.3 source ↗

For the following exercises, rewrite the expression with an exponent no higher than 1.

$\sin^{4} (3 x)$

Answer

\frac{3 + \cos (12 x) - 4 \cos (6 x)}{8}

9.3-43 §9.3 source ↗

For the following exercises, rewrite the expression with an exponent no higher than 1.

$\cos^{4} x \sin^{2} x$

Answer

\frac{2 + \cos (2 x) - 2 \cos (4 x) - \cos (6 x)}{32}

9.3-45 §9.3 source ↗

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

$\tan^{4} x$

Answer

\frac{3 + \cos (4 x) - 4 \cos (2 x)}{3 + \cos (4 x) + 4 \cos (2 x)}

9.3-47 §9.3 source ↗

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

$\sin^{2} x \cos^{2} x$

Answer

\frac{1 - \cos (4 x)}{8}

9.3-49 §9.3 source ↗

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

$\tan^{4} x \cos^{2} x$

Answer

\frac{3 + \cos (4 x) - 4 \cos (2 x)}{4 (\cos (2 x) + 1)}

9.3-51 §9.3 source ↗

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

$\cos^{2} (2 x) \sin x$

Answer

\frac{(1 + \cos (4 x)) \sin x}{2}

9.3-53 §9.3 source ↗

For the following exercises, algebraically find an equivalent function, only in terms of $\sin x$ and/or $\cos x,$ and then check the answer by graphing both functions.

$\sin (4 x)$

Answer

4 \sin x \cos x (\cos^{2} x - \sin^{2} x)

9.3-55 §9.3 source ↗

For the following exercises, prove the identities.

$\sin (2 x) =$ $\frac{2 \tan x}{1 + \tan^{2} x}$

Answer

\begin{array}{l} \frac{2 \tan x}{1 + \tan^{2} x} = \frac{\frac{2 \sin x}{\cos x}}{1 + \frac{\sin^{2} x}{\cos^{2} x}} = \frac{\frac{2 \sin x}{\cos x}}{\frac{\cos^{2} x + \sin^{2} x}{\cos^{2} x}} = \\ \frac{2 \sin x}{\cos x} . \frac{\cos^{2} x}{1} = 2 \sin x \cos x = \sin (2 x) \end{array}

9.3-57 §9.3 source ↗

For the following exercises, prove the identities.

$\tan (2 x) =$ $\frac{2 \sin x \cos x}{2 \cos^{2} x - 1}$

Answer

\frac{2 \sin x \cos x}{2 \cos^{2} x - 1} =

\frac{\sin (2 x)}{\cos (2 x)} =

\tan (2 x)

9.3-59 §9.3 source ↗

For the following exercises, prove the identities.

$\sin (3 x) =$ $3 \sin x \cos^{2} x -$ $\sin^{3} x$

Answer

\begin{matrix} \sin (x + 2 x) & = & \sin x \cos (2 x) + \sin (2 x) \cos x \\ = & \sin x (\cos^{2} x - \sin^{2} x) + 2 \sin x \cos x \cos x \\ = & \sin x \cos^{2} x - \sin^{3} x + 2 \sin x \cos^{2} x \\ = & 3 \sin x \cos^{2} x - \sin^{3} x \end{matrix}

9.3-61 §9.3 source ↗

For the following exercises, prove the identities.

$\frac{1 + \cos (2 t)}{\sin (2 t) - \cos t} =$ $\frac{2 \cos t}{2 \sin t - 1}$

Answer

\begin{matrix} \frac{1 + \cos (2 t)}{\sin (2 t) - \cos t} & = & \frac{1 + 2 \cos^{2} t - 1}{2 \sin t \cos t - \cos t} \\ = & \frac{2 \cos^{2} t}{\cos t (2 \sin t - 1)} \\ = & \frac{2 \cos t}{2 \sin t - 1} \end{matrix}

9.3-63 §9.3 source ↗

For the following exercises, prove the identities.

$\cos (16 x) =$ $(\cos^{2} (4 x) - \sin^{2} (4 x) - \sin (8 x)) (\cos^{2} (4 x) - \sin^{2} (4 x) + \sin (8 x))$

Answer

\begin{matrix} (\cos^{2} (4 x) - \sin^{2} (4 x) - \sin (8 x)) (\cos^{2} (4 x) - \sin^{2} (4 x) + \sin (8 x)) & = \\ = & (\cos (8 x) - \sin (8 x)) (\cos (8 x) + \sin (8 x)) \\ = & \cos^{2} (8 x) - \sin^{2} (8 x) \\ = & \cos (16 x) \end{matrix}

Try It #1 §9.4 source ↗

Use the product-to-sum formula to write the product as a sum or difference:

\cos (2 θ) \cos (4 θ) .

Answer

\frac{1}{2} (\cos 6 θ + \cos 2 θ)

Try It #2 §9.4 source ↗

Use the product-to-sum formula to write the product as a sum:

\sin (x + y) \cos (x - y) .

Answer

\frac{1}{2} (\sin 2 x + \sin 2 y)

Try It #3 §9.4 source ↗

Use the product-to-sum formula to evaluate

\cos \frac{11 π}{12} \cos \frac{π}{12} .

Answer

\frac{- 2 - \sqrt{3}}{4}

Try It #4 §9.4 source ↗

Use the sum-to-product formula to write the sum as a product:

\sin (3 θ) +

\sin (θ) .

Answer

2 \sin (2 θ) \cos (θ)

Try It #5 §9.4 source ↗

Verify the identity

\tan θ \cot θ -

\cos^{2} θ =

\sin^{2} θ .

Answer

\begin{matrix} \tan θ \cot θ - \cos^{2} θ & = & (\frac{\sin θ}{\cos θ}) (\frac{\cos θ}{\sin θ}) - \cos^{2} θ \\ = & 1 - \cos^{2} θ \\ = & \sin^{2} θ \end{matrix}

9.4-1 §9.4 source ↗

Starting with the product to sum formula

\sin α \cos β =

\frac{1}{2} [\sin (α + β) + \sin (α - β)],

explain how to determine the formula for

\cos α \sin β .

Answer

Substitute

α

into cosine and

β

into sine and evaluate.

9.4-3 §9.4 source ↗

Describe a situation where we would convert an equation from a sum to a product and give an example.

Answer

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example:

\frac{\sin (3 x) + \sin x}{\cos x} =

1.

When converting the numerator to a product the equation becomes:

\frac{2 \sin (2 x) \cos x}{\cos x} =

1

9.4-5 §9.4 source ↗

For the following exercises, rewrite the product as a sum or difference.

$16 \sin (16 x) \sin (11 x)$

Answer

8 (\cos (5 x) - \cos (27 x))

9.4-7 §9.4 source ↗

For the following exercises, rewrite the product as a sum or difference.

$2 \sin (5 x) \cos (3 x)$

Answer

\sin (2 x) +

\sin (8 x)

9.4-9 §9.4 source ↗

For the following exercises, rewrite the product as a sum or difference.

$\sin (- x) \sin (5 x)$

Answer

\frac{1}{2} (\cos (6 x) - \cos (4 x))

9.4-11 §9.4 source ↗

For the following exercises, rewrite the sum or difference as a product.

$\cos (6 t) +$ $\cos (4 t)$

Answer

2 \cos (5 t) \cos t

9.4-13 §9.4 source ↗

For the following exercises, rewrite the sum or difference as a product.

$\cos (7 x) +$ $\cos (- 7 x)$

Answer

2 \cos (7 x)

9.4-15 §9.4 source ↗

For the following exercises, rewrite the sum or difference as a product.

$\cos (3 x) +$ $\cos (9 x)$

Answer

2 \cos (6 x) \cos (3 x)

9.4-17 §9.4 source ↗

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\cos (45°) \cos (15°)$

Answer

\frac{1}{4} (1 + \sqrt{3})

9.4-19 §9.4 source ↗

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\sin (−345°) \sin (−15°)$

Answer

\frac{1}{4} (\sqrt{3} - 2)

9.4-21 §9.4 source ↗

For the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate exactly.

$\sin (−45°) \sin (−15°)$

Answer

\frac{1}{4} (\sqrt{3} - 1)

9.4-23 §9.4 source ↗

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

$2 \sin (100°) \sin (20°)$

Answer

\cos (80°) -

\cos (120°)

9.4-25 §9.4 source ↗

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

$\sin (213°) \cos (8°)$

Answer

\frac{1}{2} (\sin (221°) + \sin (205°))

9.4-27 §9.4 source ↗

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin (76°) +$ $\sin (14°)$

Answer

\sqrt{2} \cos (31°)

9.4-29 §9.4 source ↗

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin (101°) -$ $\sin (32°)$

Answer

2 \cos (66.5°) \sin (34.5°)

9.4-31 §9.4 source ↗

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

$\sin (−1°) +$ $\sin (−2°)$

Answer

2 \sin (−1.5°) \cos (0.5°)

9.4-33 §9.4 source ↗

For the following exercises, prove the identity.

$4 \sin (3 x) \cos (4 x) =$ $2 \sin (7 x) -$ $2 \sin x$

Answer

\begin{array}{l} 2 \sin (7 x) - 2 \sin x = 2 \sin (4 x + 3 x) - 2 \sin (4 x - 3 x) = \\ 2 (\sin (4 x) \cos (3 x) + \sin (3 x) \cos (4 x)) - 2 (\sin (4 x) \cos (3 x) - \sin (3 x) \cos (4 x)) = \\ 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) - 2 \sin (4 x) \cos (3 x) + 2 \sin (3 x) \cos (4 x)) = \\ 4 \sin (3 x) \cos (4 x) \end{array}

9.4-35 §9.4 source ↗

For the following exercises, prove the identity.

$\sin x +$ $\sin (3 x) =$ $4 \sin x \cos^{2} x$

Answer

\begin{matrix} \sin x + \sin (3 x) & = & 2 \sin (\frac{4 x}{2}) \cos (\frac{- 2 x}{2}) = \\ 2 \sin (2 x) \cos x & = & 2 (2 \sin x \cos x) \cos x = \\ 4 \sin x \cos^{2} x \end{matrix}

9.4-37 §9.4 source ↗

For the following exercises, prove the identity.

$2 \tan x \cos (3 x) =$ $\sec x (\sin (4 x) - \sin (2 x))$

Answer

\begin{array}{l} 2 \tan x \cos (3 x) = \frac{2 \sin x \cos (3 x)}{\cos x} = \frac{2 (.5 (\sin (4 x) - \sin (2 x)))}{\cos x} = \\ \frac{1}{\cos x} (\sin (4 x) - \sin (2 x)) = \sec x (\sin (4 x) - \sin (2 x)) \end{array}

9.4-39 §9.4 source ↗

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

$\cos (58°) +$ $\cos (12°)$

Answer

2 \cos (35°) \cos (23°),

1.5081

9.4-41 §9.4 source ↗

$\cos (44°) -$ $\cos (22°)$

Answer

-

2 \sin (33°) \sin (11°),

-

0.2078

9.4-43 §9.4 source ↗

$\sin (−14°) \sin (85°)$

Answer

\frac{1}{2} (\cos (99°) - \cos (71°)),

−0.2410

9.4-45 §9.4 source ↗

For the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

$\frac{\cos (10 θ) + \cos (6 θ)}{\cos (6 θ) - \cos (10 θ)} =$ $\cot (2 θ) \cot (8 θ)$

Answer

It is an identity.

9.4-47 §9.4 source ↗

$2 \cos (2 x) \cos x +$ $\sin (2 x) \sin x =$ $2 \sin x$

Answer

It is not an identity, but

2 \cos^{3} x

is.

9.4-49 §9.4 source ↗

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\frac{\sin (9 t) - \sin (3 t)}{\cos (9 t) + \cos (3 t)}$

Answer

\tan (3 t)

9.4-51 §9.4 source ↗

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\frac{\sin (3 x) - \sin x}{\sin x}$

Answer

2 \cos (2 x)

9.4-53 §9.4 source ↗

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

$\sin x \cos (15 x) -$ $\cos x \sin (15 x)$

Answer

- \sin (14 x)

9.4-55 §9.4 source ↗

For the following exercises, prove the following sum-to-product formulas.

$\cos x +$ $\cos y =$ $2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2})$

Answer

Start with $\cos x + \cos y .$ Make a substitution and let $x = α + β$ and let $y = α - β,$ so $\cos x + \cos y$ becomes $\begin{array}{l} \cos (α + β) + \cos (α - β) = \cos α \cos β - \sin α \sin β + \cos α \cos β + \sin α \sin β = \\ 2 \cos α \cos β \end{array}$

Since $x = α + β$ and $y = α - β,$ we can solve for $α$ and $β$ in terms of x and y and substitute in for $2 \cos α \cos β$ and get $2 \cos (\frac{x + y}{2}) \cos (\frac{x - y}{2}) .$

9.4-57 §9.4 source ↗

For the following exercises, prove the identity.

$\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} =$ $-$ $\cot (2 x) \cot x$

Answer

\frac{\cos (3 x) + \cos x}{\cos (3 x) - \cos x} =

\frac{2 \cos (2 x) \cos x}{- 2 \sin (2 x) \sin x} =

-

\cot (2 x) \cot x

9.4-59 §9.4 source ↗

For the following exercises, prove the identity.

$\frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} =$ $\tan y$

Answer

\begin{matrix} \frac{\cos (2 y) - \cos (4 y)}{\sin (2 y) + \sin (4 y)} & = & \frac{- 2 \sin (3 y) \sin (- y)}{2 \sin (3 y) \cos y} = \\ \frac{2 \sin (3 y) \sin (y)}{2 \sin (3 y) \cos y} & = & \tan y \end{matrix}

9.4-61 §9.4 source ↗

For the following exercises, prove the identity.

$\cos x -$ $\cos (3 x) =$ $4 \sin^{2} x \cos x$

Answer

\begin{array}{l} \cos x - \cos (3 x) = - 2 \sin (2 x) \sin (- x) = \\ 2 (2 \sin x \cos x) \sin x = 4 \sin^{2} x \cos x \end{array}

9.4-63 §9.4 source ↗

For the following exercises, prove the identity.

$\tan (\frac{π}{4} - t) =$ $\frac{1 - \tan t}{1 + \tan t}$

Answer

\tan (\frac{π}{4} - t) =

\frac{\tan (\frac{π}{4}) - \tan t}{1 + \tan (\frac{π}{4}) \tan (t)} =

\frac{1 - \tan t}{1 + \tan t}

Try It #1 §9.5 source ↗

Solve exactly the following linear equation on the interval

[0, 2 π) : 2 \sin x +

1 =

0.

Answer

x = \frac{7 π}{6}, \frac{11 π}{6}

Try It #2 §9.5 source ↗

Find all solutions for

\tan x = \sqrt{3} .

Answer

\frac{π}{3} \pm π k

Try It #3 §9.5 source ↗

Solve

\cos θ = - 0.2.

Answer

θ \approx

1.7722 \pm 2 π k

and

θ \approx

4.5110 \pm 2 π k

Try It #4 §9.5 source ↗

Solve

\sin^{2} θ =

2 \cos θ +

2,

0 \leq θ \leq 2 π .

[Hint: Make a substitution to express the equation only in terms of cosine.]

Answer

\cos θ = - 1, θ = π

Try It #5 §9.5 source ↗

Solve the quadratic equation

2 \cos^{2} θ +

\cos θ =

0.

Answer

\frac{π}{2},

\frac{2 π}{3},

\frac{4 π}{3},

\frac{3 π}{2}

9.5-1 §9.5 source ↗

Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.

Answer

There will not always be solutions to trigonometric function equations. For a basic example,

\cos (x) = −5.

9.5-3 §9.5 source ↗

When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?

Answer

If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.

9.5-5 §9.5 source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ < 2 π .$

$2 \sin θ = \sqrt{3}$

Answer

\frac{π}{3}, \frac{2 π}{3}

9.5-7 §9.5 source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ < 2 π .$

$2 \cos θ = − \sqrt{2}$

Answer

\frac{3 π}{4}, \frac{5 π}{4}

9.5-9 §9.5 source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ < 2 π .$

$\tan x = 1$

Answer

\frac{π}{4}, \frac{5 π}{4}

9.5-11 §9.5 source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ < 2 π .$

$4 \sin^{2} x - 2 = 0$

Answer

\frac{π}{4},

\frac{3 π}{4},

\frac{5 π}{4},

\frac{7 π}{4}

9.5-13 §9.5 source ↗

For the following exercises, solve exactly on $[0, 2 π) .$

$2 \cos θ = \sqrt{2}$

Answer

\frac{π}{4}, \frac{7 π}{4}

9.5-15 §9.5 source ↗

For the following exercises, solve exactly on $[0, 2 π) .$

$2 \sin θ = −1$

Answer

\frac{7 π}{6}, \frac{11 π}{6}

9.5-17 §9.5 source ↗

For the following exercises, solve exactly on $[0, 2 π) .$

$2 \sin (3 θ) = 1$

Answer

\frac{π}{18},

\frac{5 π}{18},

\frac{13 π}{18},

\frac{17 π}{18},

\frac{25 π}{18},

\frac{29 π}{18}

9.5-19 §9.5 source ↗

For the following exercises, solve exactly on $[0, 2 π) .$

$2 \cos (3 θ) = - \sqrt{2}$

Answer

\frac{3 π}{12},

\frac{5 π}{12},

\frac{11 π}{12},

\frac{13 π}{12},

\frac{19 π}{12},

\frac{21 π}{12}

9.5-21 §9.5 source ↗

For the following exercises, solve exactly on $[0, 2 π) .$

$2 \sin (π θ) = 1$

Answer

\frac{1}{6},

\frac{5}{6},

\frac{13}{6},

\frac{17}{6},

\frac{25}{6},

\frac{29}{6},

\frac{37}{6}

9.5-23 §9.5 source ↗

For the following exercises, find all exact solutions on $[0, 2 π) .$

$\sec (x) \sin (x) -$ $2 \sin (x) =$ $0$

Answer

0, \frac{π}{3}, π, \frac{5 π}{3}

9.5-25 §9.5 source ↗

For the following exercises, find all exact solutions on $[0, 2 π) .$

$2 \cos^{2} t +$ $\cos (t) =$ $1$

Answer

\frac{π}{3}, π, \frac{5 π}{3}

9.5-27 §9.5 source ↗

For the following exercises, find all exact solutions on $[0, 2 π) .$

$2 \sin (x) \cos (x) -$ $\sin (x) +$ $2 \cos (x) -$ $1 =$ $0$

Answer

\frac{π}{3}, \frac{3 π}{2}, \frac{5 π}{3}

9.5-29 §9.5 source ↗

For the following exercises, find all exact solutions on $[0, 2 π) .$

$\sec^{2} x = 1$

Answer

0, π

9.5-31 §9.5 source ↗

For the following exercises, find all exact solutions on $[0, 2 π) .$

$8 \sin^{2} (x) +$ $6 \sin (x) +$ $1 =$ $0$

Answer

π -

\sin^{- 1} (- \frac{1}{4}),

\frac{7 π}{6},

\frac{11 π}{6},

2 π +

\sin^{- 1} (- \frac{1}{4})

9.5-33 §9.5 source ↗

For the following exercises, solve with the methods shown in this section exactly on the interval $[0, 2 π) .$

$\sin (3 x) \cos (6 x) -$ $\cos (3 x) \sin (6 x) =$ $−0.9$

Answer

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

\frac{π}{3} -

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

\frac{2 π}{3} +

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

π -

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

\frac{4 π}{3} +

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

\frac{5 π}{3} -

\frac{1}{3} (\sin^{- 1} (\frac{9}{10}))

9.5-35 §9.5 source ↗

For the following exercises, solve with the methods shown in this section exactly on the interval $[0, 2 π) .$

$\cos (2 x) \cos x +$ $\sin (2 x) \sin x =$ $1$

Answer

0

9.5-37 §9.5 source ↗

For the following exercises, solve with the methods shown in this section exactly on the interval $[0, 2 π) .$

$9 \cos (2 θ) =$ $9 \cos^{2} θ -$ $4$

Answer

\frac{π}{6},

\frac{5 π}{6},

\frac{7 π}{6},

\frac{11 π}{6}

9.5-39 §9.5 source ↗

For the following exercises, solve with the methods shown in this section exactly on the interval $[0, 2 π) .$

$\cos (2 t) =$ $\sin t$

Answer

\frac{3 π}{2}, \frac{π}{6}, \frac{5 π}{6}

9.5-41 §9.5 source ↗

For the following exercises, solve exactly on the interval $[0, 2 π) .$ Use the quadratic formula if the equations do not factor.

$\tan^{2} x -$ $\sqrt{3} \tan x =$ $0$

Answer

0, \frac{π}{3}, π, \frac{4 π}{3}

9.5-43 §9.5 source ↗

For the following exercises, solve exactly on the interval $[0, 2 π) .$ Use the quadratic formula if the equations do not factor.

$\sin^{2} x -$ $2 \sin x -$ $4 =$ $0$

Answer

There are no solutions.

9.5-45 §9.5 source ↗

For the following exercises, solve exactly on the interval $[0, 2 π) .$ Use the quadratic formula if the equations do not factor.

$3 \cos^{2} x -$ $2 \cos x -$ $2 =$ $0$

Answer

\cos^{- 1} (\frac{1}{3} (1 - \sqrt{7}))

2 π -

\cos^{- 1} (\frac{1}{3} (1 - \sqrt{7}))

9.5-47 §9.5 source ↗

For the following exercises, solve exactly on the interval $[0, 2 π) .$ Use the quadratic formula if the equations do not factor.

$\tan^{2} x +$ $5 \tan x -$ $1 =$ $0$

Answer

\tan^{- 1} (\frac{1}{2} (\sqrt{29} - 5))

π +

\tan^{- 1} (\frac{1}{2} (- \sqrt{29} - 5))

π +

\tan^{- 1} (\frac{1}{2} (\sqrt{29} - 5))

2 π +

\tan^{- 1} (\frac{1}{2} (- \sqrt{29} - 5))

9.5-49 §9.5 source ↗

For the following exercises, solve exactly on the interval $[0, 2 π) .$ Use the quadratic formula if the equations do not factor.

$-$ $\tan^{2} x -$ $\tan x -$ $2 =$ $0$

Answer

There are no solutions.

9.5-51 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$\sin^{2} x +$ $\cos^{2} x =$ $0$

Answer

There are no solutions.

9.5-53 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$\cos (2 x) -$ $\cos x =$ $0$

Answer

0, \frac{2 π}{3}, \frac{4 π}{3}

9.5-55 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$1 -$ $\cos (2 x) =$ $1 +$ $\cos (2 x)$

Answer

\frac{π}{4},

\frac{3 π}{4},

\frac{5 π}{4},

\frac{7 π}{4}

9.5-57 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$10 \sin x \cos x =$ $6 \cos x$

Answer

\sin^{- 1} (\frac{3}{5}),

\frac{π}{2},

π -

\sin^{- 1} (\frac{3}{5}),

\frac{3 π}{2}

9.5-59 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$4 \cos^{2} x -$ $4 =$ $15 \cos x$

Answer

\cos^{- 1} (- \frac{1}{4}),

2 π -

\cos^{- 1} (- \frac{1}{4})

9.5-61 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$8 \cos^{2} θ =$ $3 -$ $2 \cos θ$

Answer

\frac{π}{3}

\cos^{- 1} (- \frac{3}{4})

2 π -

\cos^{- 1} (- \frac{3}{4})

\frac{5 π}{3}

9.5-63 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$12 \sin^{2} t +$ $\cos t -$ $6 =$ $0$

Answer

\cos^{- 1} (\frac{3}{4})

\cos^{- 1} (- \frac{2}{3})

2 π -

\cos^{- 1} (- \frac{2}{3})

2 π -

\cos^{- 1} (\frac{3}{4})

9.5-65 §9.5 source ↗

For the following exercises, find exact solutions on the interval $[0, 2 π) .$ Look for opportunities to use trigonometric identities.

$\cos^{3} t = \cos t$

Answer

0, \frac{π}{2}, π, \frac{3 π}{2}

9.5-67 §9.5 source ↗

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros. Use an interval of $[0, 2 π)$

$8 \cos^{2} x -$ $2 \cos x -$ $1 =$ $0$

Answer

\frac{π}{3}

\cos^{−1} (- \frac{1}{4})

2 π -

\cos^{−1} (- \frac{1}{4})

\frac{5 π}{3}

9.5-69 §9.5 source ↗

$2 \cos^{2} x -$ $\cos x +$ $15 =$ $0$

Answer

There are no solutions.

9.5-71 §9.5 source ↗

$2 \tan^{2} x +$ $7 \tan x +$ $6 =$ $0$

Answer

π + \tan^{−1} (−2)

π +

\tan^{−1} (- \frac{3}{2})

2 π +

\tan^{−1} (−2)

2 π +

\tan^{−1} (- \frac{3}{2})

9.5-73 §9.5 source ↗

For the following exercises, use a calculator to find all solutions to four decimal places.

$\sin x = 0.27$

Answer

2 π k +

0.2734,

2 π k +

2.8682

9.5-75 §9.5 source ↗

For the following exercises, use a calculator to find all solutions to four decimal places.

$\tan x = −0.34$

Answer

π k - 0.3277

9.5-77 §9.5 source ↗

For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $[0, 2 π) .$ Round to four decimal places.

$\tan^{2} x +$ $3 \tan x -$ $3 =$ $0$

Answer

0.6694,

1.8287,

3.8110,

4.9703

9.5-79 §9.5 source ↗

For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $[0, 2 π) .$ Round to four decimal places.

$\tan^{2} x -$ $\sec x =$ $1$

Answer

1.0472,

3.1416,

5.2360

9.5-81 §9.5 source ↗

For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval $[0, 2 π) .$ Round to four decimal places.

$2 \tan^{2} x +$ $9 \tan x -$ $6 =$ $0$

Answer

0.5326,

1.7648,

3.6742,

4.9064

9.5-83 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

$\csc^{2} x -$ $3 \csc x -$ $4 =$ $0$

Answer

\sin^{- 1} (\frac{1}{4}),

π -

\sin^{- 1} (\frac{1}{4}),

\frac{3 π}{2}

9.5-85 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

$\sin^{2} x (1 - \sin^{2} x) +$ $\cos^{2} x (1 - \sin^{2} x) =$ $0$

Answer

\frac{π}{2}, \frac{3 π}{2}

9.5-87 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

$\sin^{2} x -$ $1 +$ $2 \cos (2 x) -$ $\cos^{2} x =$ $1$

Answer

There are no solutions.

9.5-89 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

$\frac{\sin (2 x)}{\sec^{2} x} =$ $0$

Answer

0, \frac{π}{2}, π, \frac{3 π}{2}

9.5-91 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

$2 \cos^{2} x -$ $\sin^{2} x -$ $\cos x -$ $5 =$ $0$

Answer

There are no solutions.

9.5-93 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?

Answer

{7.2}^{\circ}

9.5-95 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?

Answer

{5.7}^{\circ}

9.5-97 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is the astronaut looking down at him from horizontal? (Hint: this is called the angle of depression.)

Answer

{82.4}^{\circ}

9.5-99 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

Issa is standing 10 meters away from a 6-meter tall building. Travis is at the top of the building looking down at Issa. At what angle is Travis looking at Issa?

Answer

{31.0}^{\circ}

9.5-101 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?

Answer

{88.7}^{\circ}

9.5-103 §9.5 source ↗

For the following exercises, find all solutions exactly to the equations on the interval $[0, 2 π) .$

A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?

Answer

{59.0}^{\circ}

9.5-105 §9.5 source ↗

For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.

A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?

Answer

{36.9}^{\circ}

9.R-1 §9.R source ↗

For the following exercises, find all solutions exactly that exist on the interval $[0, 2 π) .$

$\csc^{2} t = 3$

Answer

\sin^{- 1} (\frac{\sqrt{3}}{3})

π - \sin^{- 1} (\frac{\sqrt{3}}{3})

π + \sin^{- 1} (\frac{\sqrt{3}}{3})

2 π -

\sin^{- 1} (\frac{\sqrt{3}}{3})

9.R-3 §9.R source ↗

For the following exercises, find all solutions exactly that exist on the interval $[0, 2 π) .$

$2 \sin θ = −1$

Answer

\frac{7 π}{6}, \frac{11 π}{6}

9.R-5 §9.R source ↗

For the following exercises, find all solutions exactly that exist on the interval $[0, 2 π) .$

$9 \sin ω -$ $2 =$ $4 \sin^{2} ω$

Answer

\sin^{- 1} (\frac{1}{4}),

π -

\sin^{- 1} (\frac{1}{4})

9.R-7 §9.R source ↗

For the following exercises, use basic identities to simplify the expression.

$\sec x \cos x +$ $\cos x -$ $\frac{1}{\sec x}$

Answer

1

9.R-9 §9.R source ↗

For the following exercises, determine if the given identities are equivalent.

$\sin^{2} x +$ $\sec^{2} x -$ $1 =$ $\frac{(1 - \cos^{2} x) (1 + \cos^{2} x)}{\cos^{2} x}$

Answer

Yes

9.R-11 §9.R source ↗

For the following exercises, find the exact value.

$\tan (\frac{7 π}{12})$

Answer

- 2 - \sqrt{3}

9.R-13 §9.R source ↗

For the following exercises, find the exact value.

$\sin (70°) \cos (25°) -$ $\cos (70°) \sin (25°)$

Answer

\frac{\sqrt{2}}{2}

9.R-15 §9.R source ↗

For the following exercises, prove the identity.

$\cos (4 x) -$ $\cos (3 x) \cos x =$ $\sin^{2} x -$ $4 \cos^{2} x \sin^{2} x$

Answer

\begin{matrix} \cos (4 x) - \cos (3 x) \cos x & = & \cos (2 x + 2 x) - \cos (x + 2 x) \cos x \\ = & \cos (2 x) \cos (2 x) - \sin (2 x) \sin (2 x) - \cos x \cos (2 x) \cos x + \sin x \sin (2 x) \cos x \\ = & {(\cos^{2} x - \sin^{2} x)}^{2} - 4 \cos^{2} x \sin^{2} x - \cos^{2} x (\cos^{2} x - \sin^{2} x) + \sin x (2) \sin x \cos x \cos x \\ = & {(\cos^{2} x - \sin^{2} x)}^{2} - 4 \cos^{2} x \sin^{2} x - \cos^{2} x (\cos^{2} x - \sin^{2} x) + 2 \sin^{2} x \cos^{2} x \\ = & \cos^{4} x - 2 \cos^{2} x \sin^{2} x + \sin^{4} x - 4 \cos^{2} x \sin^{2} x - \cos^{4} x + \cos^{2} x \sin^{2} x + 2 \sin^{2} x \cos^{2} x \\ = & \sin^{4} x - 4 \cos^{2} x \sin^{2} x + \cos^{2} x \sin^{2} x \\ = & \sin^{2} x (\sin^{2} x + \cos^{2} x) - 4 \cos^{2} x \sin^{2} x \\ = & \sin^{2} x - 4 \cos^{2} x \sin^{2} x \end{matrix}

9.R-17 §9.R source ↗

For the following exercise, simplify the expression.

$\frac{\tan (\frac{1}{2} x) + \tan (\frac{1}{8} x)}{1 - \tan (\frac{1}{8} x) \tan (\frac{1}{2} x)}$

Answer

\tan (\frac{5}{8} x)

9.R-19 §9.R source ↗

For the following exercises, find the exact value.

$\tan (\sin^{- 1} (0) + \sin^{- 1} (\frac{1}{2}))$

Answer

\frac{\sqrt{3}}{3}

9.R-21 §9.R source ↗

For the following exercises, find the exact value.

Find $\sin (2 θ),$ $\cos (2 θ),$ and $\tan (2 θ)$ given $\sec θ = - \frac{5}{3}$ and $θ$ is in the interval $[\frac{π}{2}, π] .$

Answer

-

\frac{24}{25},

-

\frac{7}{25},

\frac{24}{7}

9.R-23 §9.R source ↗

For the following exercises, find the exact value.

$\sec (\frac{3 π}{8})$

Answer

\sqrt{2 (2 + \sqrt{2})}

9.R-25 §9.R source ↗

For the following exercises, use Figure 1 to find the desired quantities.

Figure 1

$\sin (\frac{β}{2}),$ $\cos (\frac{β}{2}),$ $\tan (\frac{β}{2}),$ $\sin (\frac{α}{2}),$ $\cos (\frac{α}{2}),$ $and \tan (\frac{α}{2})$

Answer

\frac{\sqrt{2}}{10},

\frac{7 \sqrt{2}}{10},

\frac{1}{7},

\frac{3}{5},

\frac{4}{5},

\frac{3}{4}

9.R-27 §9.R source ↗

For the following exercises, prove the identity.

$\cot x \cos (2 x) =$ $-$ $\sin (2 x) +$ $\cot x$

Answer

\begin{matrix} \cot x \cos (2 x) & = & \cot x (1 - 2 \sin^{2} x) \\ = & \cot x - \frac{\cos x}{\sin x} (2) \sin^{2} x \\ = & - 2 \sin x \cos x + \cot x \\ = & - \sin (2 x) + \cot x \end{matrix}

9.R-29 §9.R source ↗

For the following exercises, rewrite the expression with no powers.

$\tan^{2} x \sin^{3} x$

Answer

\frac{10 \sin x - 5 \sin (3 x) + \sin (5 x)}{8 (\cos (2 x) + 1)}

9.R-31 §9.R source ↗

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

$2 \sin (\frac{2 π}{3}) \sin (\frac{5 π}{6})$

Answer

\frac{\sqrt{3}}{2}

9.R-33 §9.R source ↗

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

$\sin (\frac{π}{12}) -$ $\sin (\frac{7 π}{12})$

Answer

- \frac{\sqrt{2}}{2}

9.R-35 §9.R source ↗

For the following exercises, change the functions from a product to a sum or a sum to a product.

$\sin (9 x) \cos (3 x)$

Answer

\frac{1}{2} (\sin (6 x) + \sin (12 x))

9.R-37 §9.R source ↗

For the following exercises, change the functions from a product to a sum or a sum to a product.

$\sin (11 x) +$ $\sin (2 x)$

Answer

2 \sin (\frac{13}{2} x) \cos (\frac{9}{2} x)

9.R-39 §9.R source ↗

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

$\tan x + 1 = 0$

Answer

\frac{3 π}{4}, \frac{7 π}{4}

9.R-41 §9.R source ↗

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

$2 \sin^{2} x -$ $\sin x =$ $0$

Answer

0, \frac{π}{6}, \frac{5 π}{6}, π

9.R-43 §9.R source ↗

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

$2 \sin^{2} x +$ $5 \sin x +$ $3 =$ $0$

Answer

\frac{3 π}{2}

9.R-45 §9.R source ↗

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

$\frac{1}{\sec^{2} x} +$ $2 +$ $\sin^{2} x +$ $4 \cos^{2} x =$ $0$

Answer

No solution

9.R-47 §9.R source ↗

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $[0, 2 π) .$ Round to four decimal places.

$\csc^{2} x -$ $3 \csc x -$ $4 =$ $0$

Answer

0.2527,

2.8889,

4.7124

9.R-49 §9.R source ↗

For the following exercises, graph each side of the equation to find the approximate solutions on the interval $[0, 2 π) .$

$\sec^{2} x -$ $2 \sec x =$ $15$

Answer

1.3694,

1.9106,

4.3726,

4.9137

9.T-1 §9.T source ↗

For the following exercises, simplify the given expression.

$\cos (- x) \sin x \cot x +$ $\sin^{2} x$

Answer

9.T-3 §9.T source ↗

For the following exercises, simplify the given expression.

$\csc (θ) \cot (θ) (\sec^{2} θ - 1)$

Answer

\sec (θ)

9.T-5 §9.T source ↗

For the following exercises, find the exact value.

$\cos (\frac{7 π}{12})$

Answer

\frac{\sqrt{2} - \sqrt{6}}{4}

9.T-7 §9.T source ↗

For the following exercises, find the exact value.

$\tan (\sin^{- 1} (\frac{\sqrt{2}}{2}) + \tan^{- 1} \sqrt{3})$

Answer

- 2 - \sqrt{3}

9.T-9 §9.T source ↗

For the following exercises, find the exact value.

$\cos (\frac{4 π}{3} + θ)$

Answer

-

\frac{1}{2} \cos θ +

\frac{\sqrt{3}}{2} \sin θ

9.T-11 §9.T source ↗

For the following exercises, simplify each expression. Do not evaluate.

$\cos^{2} (32°) \tan^{2} (32°)$

Answer

\frac{1 - \cos (64^{\circ})}{2}

9.T-13 §9.T source ↗

For the following exercises, find all exact solutions to the equation on $[0, 2 π) .$

$\cos^{2} x -$ $\sin^{2} x -$ $1 =$ $0$

Answer

0, π

9.T-15 §9.T source ↗

For the following exercises, find all exact solutions to the equation on $[0, 2 π) .$

$\cos (2 x) +$ $\sin^{2} x =$ $0$

Answer

\frac{π}{2}, \frac{3 π}{2}

9.T-17 §9.T source ↗

For the following exercises, find all exact solutions to the equation on $[0, 2 π) .$

Rewrite the expression as a product instead of a sum: $\cos (2 x) +$ $\cos (- 8 x) .$

Answer

2 cos (3 x) cos (5 x)

9.T-19 §9.T source ↗

For the following exercise, rewrite the sum or difference as a product.

$2 (\sin (8 θ) - \sin (4 θ))$

Answer

4 \sin (2 θ) \cos (6 θ)

9.T-21 §9.T source ↗

For the following exercise, rewrite the sum or difference as a product.

Find the solutions of $\sec^{2} x -$ $2 \sec x =$ $15$ on the interval $[0, 2 π)$ algebraically; then graph both sides of the equation to determine the answer.

Answer

x = {cos}^{–1} (\frac{1}{5})

9.T-23 §9.T source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ \leq π$

$\sqrt{3} \cot (y) = 1$

Answer

\frac{π}{3}

9.T-25 §9.T source ↗

For the following exercises, find all solutions exactly on the interval $0 \leq θ \leq π$

Find $\sin (\frac{θ}{2}),$ $\cos (\frac{θ}{2}),$ and $\tan (\frac{θ}{2})$ given $\cos θ = \frac{7}{25}$ and $θ$ is in quadrant IV.

Answer

\frac{3}{5}, - \frac{4}{5}, - \frac{3}{4}

9.T-27 §9.T source ↗

For the following exercises, prove the identity.

$\tan^{3} x -$ $\tan x \sec^{2} x =$ $\tan (- x)$

Answer

\begin{matrix} {tan}^{3} x - tan x {sec}^{2} x & = & tan x ({tan}^{2} x - {sec}^{2} x) \\ = & tan x ({tan}^{2} x - (1 + {tan}^{2} x)) \\ = & tan x ({tan}^{2} x - 1 - {tan}^{2} x) \\ = & - tan x = tan (- x) = tan (- x) \end{matrix}

9.T-29 §9.T source ↗

For the following exercises, prove the identity.

$\frac{\sin (2 x)}{\sin x} -$ $\frac{\cos (2 x)}{\cos x} =$ $\sec x$

Answer

\begin{matrix} \frac{sin (2 x)}{sin x} - \frac{cos (2 x)}{cos x} & = & \frac{2 sin x cos x}{sin x} - \frac{2 {cos}^{2} x - 1}{cos x} \\ = & 2 cos x - 2 cos x + \frac{1}{cos x} \\ = & \frac{1}{cos x} = sec x = sec x \end{matrix}

9.T-31 §9.T source ↗

For the following exercises, prove the identity.

The displacement $h (t)$ in centimeters of a mass suspended by a spring is modeled by the function $h (t) =$ $\frac{1}{4} \sin (120 π t),$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Answer

Amplitude:

\frac{1}{4}

, period:

\frac{1}{60}

, frequency: 60 Hz

9.T-33 §9.T source ↗

For the following exercises, prove the identity.

Two frequencies of sound are played on an instrument governed by the equation $n (t) =$ $8 \cos (20 π t) \cos (1000 π t) .$ What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

Answer

Amplitude: 8, fast period:

\frac{1}{500}

, fast frequency: 500 Hz, slow period:

\frac{1}{10}

, slow frequency: 10 Hz

9.T-35 §9.T source ↗

For the following exercises, prove the identity.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring $t$ seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

Answer

D (t) =

20 (0.9086)^{t} cos (4 π t)

, 31 second

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Problems and answers are adapted from Algebra and Trigonometry 2e by Jay Abramson, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.