Make the trigonometric substitution \[x = a \tan \theta \text { for } \frac { - \pi } { 2 } 0. Use fundamental identities to simplify the resulting expression. $\frac { 1 } { x ^ { 2 } + a ^ { 2 } }$

A) $a ^ { 2 } \sec ^ { 2 } \theta \csc \theta$ B) $\frac { 1 } { a } \cos ^ { 2 } \theta$ C) $\frac { 1 } { a } \cos \theta$ D) $\frac { 1 } { a ^ { 2 } } \cos ^ { 2 } \theta$ E) $a \sec \theta$ A) $a ^ { 2 } \sec ^ { 2 } \theta \csc \theta$ B) $\frac { 1 } { a } \cos ^ { 2 } \theta$ C) $\frac { 1 } { a } \cos \theta$ D) $\frac { 1 } { a ^ { 2 } } \cos ^ { 2 } \theta$ E) $a \sec \theta$

Exam 6: Analytic Trigonometry

Make the trigonometric substitution $x = a \tan \theta \text { for } \frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$ and a > 0. Use fundamental identities to simplify the resulting expression. $\frac { 1 } { x ^ { 2 } + a ^ { 2 } }$

(Multiple Choice)

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Question 21

If an earthquake has a total horizontal displacement of S meters along its fault line, then the horizontal movement M of a point on the surface of Earth d kilometers from the fault line can be estimated using the formula $M = \frac { S } { 2 } \left( 1 - \frac { 2 } { \pi } \tan ^ { - 1 } \frac { d } { D } \right)$ where D is the depth (in kilometers) below the surface of the focal point of the earthquake. For the San Francisco earthquake of 1906, S was 4 meters and D was 3.08 kilometers. Approximate M for d = 6 kilometers. Round the answer to the nearest hundredth.

(Multiple Choice)

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Question 22

Verify the identity. $\sin ^ { 3 } t + \cos ^ { 3 } t = ( 1 - \sin t \cos t ) ( \sin t + \cos t )$

(Multiple Choice)

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Question 23

If $\tan \alpha = - \frac { 7 } { 24 }$ and $\beta = \frac { 3 } { 4 }$ for a second-quadrant angle $\alpha$ and a third-quadrant angle $\beta$ , find $\cos ( \alpha + \beta )$

(Multiple Choice)

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Question 24

Use the graph of f to find the simplest expression g(x) such that the equation $f ( x ) = g ( x )$ is an identity. $f ( x ) = \frac { \sin 2 x + \sin x } { \cos 2 x + \cos x + 1 }$

(Multiple Choice)

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Question 25

Make the trigonometric substitution $x = a \tan \theta \text { for } \frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$ and a > 0. Use fundamental identities to simplify the resulting expression. $\frac { 1 } { x ^ { 2 } + a ^ { 2 } }$

Verify the identity. $\sin ^ { 3 } t + \cos ^ { 3 } t = ( 1 - \sin t \cos t ) ( \sin t + \cos t )$

If $\tan \alpha = - \frac { 7 } { 24 }$ and $\beta = \frac { 3 } { 4 }$ for a second-quadrant angle $\alpha$ and a third-quadrant angle $\beta$ , find $\cos ( \alpha + \beta )$

Use the graph of f to find the simplest expression g(x) such that the equation $f ( x ) = g ( x )$ is an identity. $f ( x ) = \frac { \sin 2 x + \sin x } { \cos 2 x + \cos x + 1 }$

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Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

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Exam 6: Analytic Trigonometry

Verify the identity. sin⁡3t+cos⁡3t=(1−sin⁡tcos⁡t)(sin⁡t+cos⁡t)\sin ^ { 3 } t + \cos ^ { 3 } t = ( 1 - \sin t \cos t ) ( \sin t + \cos t )sin3t+cos3t=(1−sintcost)(sint+cost)

If tan⁡α=−724\tan \alpha = - \frac { 7 } { 24 }tanα=−247​ and β=34\beta = \frac { 3 } { 4 }β=43​ for a second-quadrant angle α\alphaα and a third-quadrant angle β\betaβ , find cos⁡(α+β)\cos ( \alpha + \beta )cos(α+β)

Use the graph of f to find the simplest expression g(x) such that the equation f(x)=g(x)f ( x ) = g ( x )f(x)=g(x) is an identity. f(x)=sin⁡2x+sin⁡xcos⁡2x+cos⁡x+1f ( x ) = \frac { \sin 2 x + \sin x } { \cos 2 x + \cos x + 1 }f(x)=cos2x+cosx+1sin2x+sinx​

Topics From Algebra

Functions and Graphs

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytical Geometry

Filters

Verify the identity. $\sin ^ { 3 } t + \cos ^ { 3 } t = ( 1 - \sin t \cos t ) ( \sin t + \cos t )$

If $\tan \alpha = - \frac { 7 } { 24 }$ and $\beta = \frac { 3 } { 4 }$ for a second-quadrant angle $\alpha$ and a third-quadrant angle $\beta$ , find $\cos ( \alpha + \beta )$

Use the graph of f to find the simplest expression g(x) such that the equation $f ( x ) = g ( x )$ is an identity. $f ( x ) = \frac { \sin 2 x + \sin x } { \cos 2 x + \cos x + 1 }$