Exam 5: Analytic Trigonometry

Determine which of the following are trigonometric identities. 6 I. $\frac { \cos ( 4 x ) + \cos ( 2 x ) } { 2 \cot ( 3 x ) } = \cot ( x )$ II. $\frac { \cos ( 4 x ) + \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = \cot ( 2 x )$ III. $\frac { \cos ( 6 x ) + \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = \cot ( 3 x )$

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(Multiple Choice)

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Question 1

Correct Answer:

Verified

A

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Free

(Multiple Choice)

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Question 2

Correct Answer:

Verified

C

Verify the given identity. $\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$

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(Essay)

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Question 3

Correct Answer:

Verified

$\begin{aligned}\frac { \cos u - \cos v } { \cos u + \cos v } & = \frac { - 2 \sin \left( \frac { u + v } { 2 } \right) \sin \left( \frac { u - v } { 2 } \right) } { 2 \cos \left( \frac { u + v } { 2 } \right) \cos \left( \frac { u - v } { 2 } \right) } \\& = - \tan \left( \frac { u + v } { 2 } \right) \tan \left( \frac { u - v } { 2 } \right) \\& = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )\end{aligned}$

Use the product-to-sum formula to write the given product as a sum or difference. $8 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }$

(Multiple Choice)

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Question 4

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

(Essay)

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Question 5

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\sin \alpha ( \csc \alpha - \sin \alpha )$

(Multiple Choice)

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Question 6

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 11 } { 61 }$ and $\cos v = \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant IV.)

(Multiple Choice)

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Question 7

Determine which of the following are trigonometric identities. I. $\frac { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 0$ II. $\frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } + \frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } = 1$ III. $\frac { \cos ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) \sin ( \mathrm { s } ) } = \cos ( \mathrm { s } ) + \sin ( \mathrm { t } )$

(Multiple Choice)

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Question 8

If $x = 10 \sin \theta$ , use trigonometric substitution to write $\sqrt { 100 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .

(Multiple Choice)

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Question 9

Use the sum-to-product formulas to write the given expression as a product. $\cos 8 \theta - \cos 6 \theta$

(Multiple Choice)

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Question 10

Use the half-angle formulas to determine the exact value of the following. $\cos \left( - 22.5 ^ { \circ } \right)$

(Multiple Choice)

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Question 11

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

(Multiple Choice)

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Question 12

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

(Multiple Choice)

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Question 13

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

(Multiple Choice)

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Question 14

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

(Multiple Choice)

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Question 15

Write the given expression as an algebraic expression. $\cos ( \arccos x - \arcsin x )$

(Multiple Choice)

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Question 16

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ .

(Multiple Choice)

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Question 17

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

(Essay)

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Question 18

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

(Multiple Choice)

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Question 19

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$ . (Both $u$ and $v$ are in Quadrant IV.)

(Multiple Choice)

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Question 20

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Verify the given identity. $\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$

Use the product-to-sum formula to write the given product as a sum or difference. $8 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }$

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\sin \alpha ( \csc \alpha - \sin \alpha )$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 11 } { 61 }$ and $\cos v = \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant IV.)

If $x = 10 \sin \theta$ , use trigonometric substitution to write $\sqrt { 100 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .

Use the sum-to-product formulas to write the given expression as a product. $\cos 8 \theta - \cos 6 \theta$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( - 22.5 ^ { \circ } \right)$

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

Write the given expression as an algebraic expression. $\cos ( \arccos x - \arcsin x )$

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ .

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Functions and Their Graphs

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Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

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Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

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Exam 5: Analytic Trigonometry

Expand the expression below and use fundamental trigonometric identities to simplify. (sin⁡(ω)+cos⁡(ω))2( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }(sin(ω)+cos(ω))2

Verify the given identity. cos⁡u−cos⁡vcos⁡u+cos⁡v=−tan⁡12(u+v)tan⁡12(u−v)\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )cosu+cosvcosu−cosv​=−tan21​(u+v)tan21​(u−v)

Use the product-to-sum formula to write the given product as a sum or difference. 8sin⁡π8sin⁡π88 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }8sin8π​sin8π​

Verify the identity shown below. tan⁡α+cot⁡βtan⁡αcot⁡β=tan⁡β+cot⁡α\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alphatanαcotβtanα+cotβ​=tanβ+cotα

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. sin⁡α(csc⁡α−sin⁡α)\sin \alpha ( \csc \alpha - \sin \alpha )sinα(cscα−sinα)

Find the exact value of tan⁡(u+v)\tan ( u + v )tan(u+v) given that sin⁡u=−1161\sin u = - \frac { 11 } { 61 }sinu=−6111​ and cos⁡v=4041\cos v = \frac { 40 } { 41 }cosv=4140​ . (Both uuu and vvv are in Quadrant IV.)

If x=10sin⁡θx = 10 \sin \thetax=10sinθ , use trigonometric substitution to write 100−x2\sqrt { 100 - x ^ { 2 } }100−x2​ as a trigonometric function of θ\thetaθ , where −π2<θ<π2- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }−2π​<θ<2π​ .

Use the sum-to-product formulas to write the given expression as a product. cos⁡8θ−cos⁡6θ\cos 8 \theta - \cos 6 \thetacos8θ−cos6θ

Use the half-angle formulas to determine the exact value of the following. cos⁡(−22.5∘)\cos \left( - 22.5 ^ { \circ } \right)cos(−22.5∘)

Expand the expression below and use fundamental trigonometric identities to simplify. (sin⁡(ω)+cos⁡(ω))2( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }(sin(ω)+cos(ω))2

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. (sin⁡x+cos⁡x)(sin⁡x−cos⁡x)( \sin x + \cos x ) ( \sin x - \cos x )(sinx+cosx)(sinx−cosx)

Write the given expression as the sine of an angle. sin⁡85∘cos⁡50∘+sin⁡50∘cos⁡85∘\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }sin85∘cos50∘+sin50∘cos85∘

Use the half-angle formulas to determine the exact value of the following. cos⁡(22.5∘)\cos \left( 22.5 ^ { \circ } \right)cos(22.5∘)

Write the given expression as an algebraic expression. cos⁡(arccos⁡x−arcsin⁡x)\cos ( \arccos x - \arcsin x )cos(arccosx−arcsinx)

Find the exact solutions of the given equation in the interval [0,2π)[ 0,2 \pi )[0,2π) .

Verify the identity shown below. tan⁡θ+1sec⁡θ+csc⁡θ=sin⁡θ\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \thetasecθ+cscθtanθ+1​=sinθ

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. sin⁡x+1+1sin⁡x−1\sin x + 1 + \frac { 1 } { \sin x - 1 }sinx+1+sinx−11​

Find the exact value of tan⁡(u+v)\tan ( u + v )tan(u+v) given that sin⁡u=−35\sin u = - \frac { 3 } { 5 }sinu=−53​ and cos⁡v=2425\cos v = \frac { 24 } { 25 }cosv=2524​ . (Both uuu and vvv are in Quadrant IV.)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Verify the given identity. $\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$

Use the product-to-sum formula to write the given product as a sum or difference. $8 \sin \frac { \pi } { 8 } \sin \frac { \pi } { 8 }$

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\sin \alpha ( \csc \alpha - \sin \alpha )$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 11 } { 61 }$ and $\cos v = \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant IV.)

If $x = 10 \sin \theta$ , use trigonometric substitution to write $\sqrt { 100 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .

Use the sum-to-product formulas to write the given expression as a product. $\cos 8 \theta - \cos 6 \theta$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( - 22.5 ^ { \circ } \right)$

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

Write the given expression as an algebraic expression. $\cos ( \arccos x - \arcsin x )$

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ .

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 3 } { 5 }$ and $\cos v = \frac { 24 } { 25 }$ . (Both $u$ and $v$ are in Quadrant IV.)