Question 1

Solve the problem.
-If $\tan \theta = \frac { 7 } { 24 }$, and $\theta$ terminates in quadrant III, then find $\cos 2 \theta$.

Accepted Answer

A)  $\frac { 527 } { 625 }$ 
B)  $- \frac { 527 } { 625 }$ 
C)  $\frac { 336 } { 625 }$ 
D)  $- \frac { 336 } { 625 }$ 
A)  $\frac { 527 } { 625 }$ 
B)  $- \frac { 527 } { 625 }$ 
C)  $\frac { 336 } { 625 }$ 
D)  $- \frac { 336 } { 625 }$

Question 2

Establish the identity.
-$$\frac { \sin x } { 1 - \cos x } + \frac { \sin x } { 1 + \cos x } = 2 \csc x$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { \sin x }...

Question 3

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = \cos ( x - 3 ) + 4$$

Accepted Answer

A)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$ 
B)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 3,5 ]$ 
D)  Domain of $f : [ - 3,3 ]$

Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
A)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$ 
B)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 3,5 ]$ 
D)  Domain of $f : [ - 3,3 ]$

Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Question 4

Solve the equation on the interval $$0 \leq \theta < 2 \pi \text {. }$$
-$\sin \theta + \sqrt { 3 } \cos \theta = - 1$

Accepted Answer

A)  $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 }$ 
B)  $0 , \frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
D)  $\frac { 3 \pi } { 2 } , \frac { \pi } { 6 }$ 
A)  $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 6 }$ 
B)  $0 , \frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
D)  $\frac { 3 \pi } { 2 } , \frac { \pi } { 6 }$

Question 5

Solve the problem.
-The formula
$$\mathrm { D } = 24 \left[ 1 - \frac { \cos ^ { - 1 } ( \tan i \tan \theta ) } { \pi } \right]$$
can be used to approximate the number of hours of daylight when the declination of the sun is $i ^ { \circ }$ at a location $\theta ^ { \circ }$ latitude for any date between the vernal equinox and autumnal equinox. To use this formula, $\cos ^ { - 1 } \left( \tan ^ { i } \tan \theta \right.$ ) must be expressed in radians. Approximate the number of hours of daylight in Flagstaff, Arizona, ( $35 ^ { \circ } 13 ^ { \prime }$ north latitude) for summer solstice $\left( \mathrm { i } = 23.5 ^ { \circ } \right)$.

Accepted Answer

The answer of Solve the problem.
-The formula
\[\mathrm { D }...

Question 6

Complete the identity.
-$\cos \left( \sin ^ { - 1 } v \right) = ?$

Accepted Answer

A)  $\sqrt { v ^ { 2 } + 1 }$ 
B)  $\sqrt { 1 - \mathrm { v } ^ { 2 } }$ 
C)  $\frac { \sqrt { v ^ { 2 } + 1 } } { v }$ 
D)  $\sqrt { v ^ { 2 } - 1 }$ 
A)  $\sqrt { v ^ { 2 } + 1 }$ 
B)  $\sqrt { 1 - \mathrm { v } ^ { 2 } }$ 
C)  $\frac { \sqrt { v ^ { 2 } + 1 } } { v }$ 
D)  $\sqrt { v ^ { 2 } - 1 }$

Question 7

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = \frac { 1 } { 4 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ Find $\sin \frac { \theta } { 2 }$

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$

Question 8

Solve the equation on the interval $0 \leq \theta < 2 \pi .$
-$2 \sin ^ { 2 } \theta = \sin \theta$

Accepted Answer

A)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
A)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 } , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
D)  $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$

Question 9

Find the exact value of the composition.
-$$\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$$

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$

Question 10

Complete the identity.
-$\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) =$ ?

Accepted Answer

A)  $2 \cos ( 2 \theta )$ 
B)  $\cos ( 3 \theta )$ 
C)  1 
D)  $\sec ( 2 \theta )$ 
A)  $2 \cos ( 2 \theta )$ 
B)  $\cos ( 3 \theta )$ 
C)  1 
D)  $\sec ( 2 \theta )$

Question 11

Complete the identity.
-$$\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta = ?$$

Accepted Answer

A)  $\tan ^ { 2 } \theta$ 
B)  $\csc ^ { 2 } \theta$ 
C)  1 
D)  $\sec ^ { 2 } \theta$ 
A)  $\tan ^ { 2 } \theta$ 
B)  $\csc ^ { 2 } \theta$ 
C)  1 
D)  $\sec ^ { 2 } \theta$

Question 12

Find the exact value of the expression. Do not use a calculator.
-$\cos \left[ \cos ^ { - 1 } ( - 0.9372 ) \right]$

Accepted Answer

A)  -0.4686 
B)  -0.9372 
C)  0.9372 
D)  0.4686 
A)  -0.4686 
B)  -0.9372 
C)  0.9372 
D)  0.4686

Question 13

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sec \theta = - \frac { 5 } { 4 } , \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \frac { \theta } { 2 }$.

Accepted Answer

A)  $- \frac { \sqrt { 10 } } { 10 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C)  $\frac { 3 \sqrt { 10 } } { 10 }$ 
D)  $- \frac { 3 } { 10 }$ 
A)  $- \frac { \sqrt { 10 } } { 10 }$ 
B)  $\frac { \sqrt { 10 } } { 10 }$ 
C)  $\frac { 3 \sqrt { 10 } } { 10 }$ 
D)  $- \frac { 3 } { 10 }$

Question 14

Find the exact value of the expression.
-$$\frac { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } } { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } }$$

Accepted Answer

A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $- \sqrt { 3 }$ 
D)  $- \frac { \sqrt { 3 } } { 3 }$ 
A)  $\sqrt { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $- \sqrt { 3 }$ 
D)  $- \frac { \sqrt { 3 } } { 3 }$

Question 15

Complete the identity.
-$\frac { \csc \theta \left( \sin ^ { 2 } \theta + \cos ^ { 2 } \theta \tan \theta \right) } { \sin \theta + \cos \theta } = ?$

Accepted Answer

A)  $\csc ^ { 2 } \theta + \tan ^ { 2 } \theta$ 
B)  $\csc \theta \tan \theta$ 
C)  1 
D)  $\sin \theta + \cos \theta$ 
A)  $\csc ^ { 2 } \theta + \tan ^ { 2 } \theta$ 
B)  $\csc \theta \tan \theta$ 
C)  1 
D)  $\sin \theta + \cos \theta$

Question 16

Find the exact value of the expression.
-$\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

Accepted Answer

A)  1 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 24 } { 25 }$ 
A)  1 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
D)  $\frac { 24 } { 25 }$

Question 17

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = 6 \tan x + 10$$

Accepted Answer

A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 4,16 ]$ 
B)  Domain of f: $x \neq \frac { ( 2 k + 1 ) \pi } { 2 } ; k$ an integer
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $\mathrm { f } : ( * , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : \mathrm { x } \neq \frac { ( 2 \mathrm { k } + 1 ) \pi } { 2 } ; \mathrm { k }$ an integer 
D)  Domain of $f : x \neq \frac { ( 2 k + 1 ) \pi } { 2 } ; k$ an integer

Domain of $f ^ { - 1 } : [ 4,16 ]$ 
A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 4,16 ]$ 
B)  Domain of f: $x \neq \frac { ( 2 k + 1 ) \pi } { 2 } ; k$ an integer
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $\mathrm { f } : ( * , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : \mathrm { x } \neq \frac { ( 2 \mathrm { k } + 1 ) \pi } { 2 } ; \mathrm { k }$ an integer 
D)  Domain of $f : x \neq \frac { ( 2 k + 1 ) \pi } { 2 } ; k$ an integer

Domain of $f ^ { - 1 } : [ 4,16 ]$

Question 18

Write the trigonometric expression as an algebraic expression in u.
-$\cos \left( \tan ^ { - 1 } \mathrm { u } \right)$

Accepted Answer

A)  $\mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 1 } }$ 
C)  $\frac { 1 } { \sqrt { \mathrm { u } ^ { 2 } + 1 } }$ 
D)  $\frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$ 
A)  $\mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 }$ 
B)  $\frac { \mathrm { u } } { \sqrt { \mathrm { u } ^ { 2 } + 1 } }$ 
C)  $\frac { 1 } { \sqrt { \mathrm { u } ^ { 2 } + 1 } }$ 
D)  $\frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$

Question 19

Complete the identity.
-$\tan \alpha + \cot \beta = ?$

Accepted Answer

A)  $\frac { \sin ( \alpha - \beta ) } { \cos \alpha \sin \beta }$ 
B)  $\frac { \cos ( \alpha - \beta ) } { \cos \alpha \sin \beta }$ 
C)  $\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta }$ 
D)  $\frac { \sin ( \alpha + \beta ) } { \cos \alpha \sin \beta }$ 
A)  $\frac { \sin ( \alpha - \beta ) } { \cos \alpha \sin \beta }$ 
B)  $\frac { \cos ( \alpha - \beta ) } { \cos \alpha \sin \beta }$ 
C)  $\frac { \cos ( \alpha + \beta ) } { \cos \alpha \sin \beta }$ 
D)  $\frac { \sin ( \alpha + \beta ) } { \cos \alpha \sin \beta }$

Question 20

Establish the identity.
-$$\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$$

Accepted Answer

The answer of Establish the identity.
-\[\sin ^ { 3 }...

Solve the problem. -If $\tan \theta = \frac { 7 } { 24 }$ , and $\theta$ terminates in quadrant III, then find $\cos 2 \theta$ .

Establish the identity. - $\frac { \sin x } { 1 - \cos x } + \frac { \sin x } { 1 + \cos x } = 2 \csc x$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = \cos ( x - 3 ) + 4$

Solve the equation on the interval $0 \leq \theta < 2 \pi \text {. }$ - $\sin \theta + \sqrt { 3 } \cos \theta = - 1$

Complete the identity. - $\cos \left( \sin ^ { - 1 } v \right) = ?$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sin \theta = \frac { 1 } { 4 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ Find $\sin \frac { \theta } { 2 }$

Solve the equation on the interval $0 \leq \theta < 2 \pi .$ - $2 \sin ^ { 2 } \theta = \sin \theta$

Find the exact value of the composition. - $\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

Complete the identity. - $\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) =$ ?

Complete the identity. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta = ?$

Find the exact value of the expression. Do not use a calculator. - $\cos \left[ \cos ^ { - 1 } ( - 0.9372 ) \right]$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sec \theta = - \frac { 5 } { 4 } , \frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \frac { \theta } { 2 }$ .

Find the exact value of the expression. - $\frac { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } } { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } }$

Complete the identity. - $\frac { \csc \theta \left( \sin ^ { 2 } \theta + \cos ^ { 2 } \theta \tan \theta \right) } { \sin \theta + \cos \theta } = ?$

Find the exact value of the expression. - $\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = 6 \tan x + 10$

Write the trigonometric expression as an algebraic expression in u. - $\cos \left( \tan ^ { - 1 } \mathrm { u } \right)$

Complete the identity. - $\tan \alpha + \cot \beta = ?$

Establish the identity. - $\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

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Exam 6: Analytic Trigonometry

Solve the problem. -If tan⁡θ=724\tan \theta = \frac { 7 } { 24 }tanθ=247​ , and θ\thetaθ terminates in quadrant III, then find cos⁡2θ\cos 2 \thetacos2θ .

Establish the identity. - sin⁡x1−cos⁡x+sin⁡x1+cos⁡x=2csc⁡x\frac { \sin x } { 1 - \cos x } + \frac { \sin x } { 1 + \cos x } = 2 \csc x1−cosxsinx​+1+cosxsinx​=2cscx

Find the domain of the function f and of its inverse function f−1\mathrm { f } ^ { - 1 }f−1 . - f(x)=cos⁡(x−3)+4f ( x ) = \cos ( x - 3 ) + 4f(x)=cos(x−3)+4

Solve the equation on the interval 0≤θ<2π. 0 \leq \theta < 2 \pi \text {. }0≤θ<2π. - sin⁡θ+3cos⁡θ=−1\sin \theta + \sqrt { 3 } \cos \theta = - 1sinθ+3​cosθ=−1

Complete the identity. - cos⁡(sin⁡−1v)=?\cos \left( \sin ^ { - 1 } v \right) = ?cos(sin−1v)=?

Solve the equation on the interval 0≤θ<2π.0 \leq \theta < 2 \pi .0≤θ<2π. - 2sin⁡2θ=sin⁡θ2 \sin ^ { 2 } \theta = \sin \theta2sin2θ=sinθ

Find the exact value of the composition. - cos⁡−1(sin⁡7π6)\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)cos−1(sin67π​)

Complete the identity. - sin⁡(2θ)tan⁡θ+cos⁡(2θ)=\sin ( 2 \theta ) \tan \theta + \cos ( 2 \theta ) =sin(2θ)tanθ+cos(2θ)= ?

Complete the identity. - 1cot⁡2θ+sec⁡θcos⁡θ=?\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta = ?cot2θ1​+secθcosθ=?

Find the exact value of the expression. Do not use a calculator. - cos⁡[cos⁡−1(−0.9372)]\cos \left[ \cos ^ { - 1 } ( - 0.9372 ) \right]cos[cos−1(−0.9372)]

Find the exact value of the expression. - 1−tan⁡80∘tan⁡70∘tan⁡80∘+tan⁡70∘\frac { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } } { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } }tan80∘+tan70∘1−tan80∘tan70∘​

Complete the identity. - csc⁡θ(sin⁡2θ+cos⁡2θtan⁡θ)sin⁡θ+cos⁡θ=?\frac { \csc \theta \left( \sin ^ { 2 } \theta + \cos ^ { 2 } \theta \tan \theta \right) } { \sin \theta + \cos \theta } = ?sinθ+cosθcscθ(sin2θ+cos2θtanθ)​=?

Find the exact value of the expression. - cos⁡(tan⁡−143−sin⁡−135)\cos \left( \tan ^ { - 1 } \frac { 4 } { 3 } - \sin ^ { - 1 } \frac { 3 } { 5 } \right)cos(tan−134​−sin−153​)

Find the domain of the function f and of its inverse function f−1\mathrm { f } ^ { - 1 }f−1 . - f(x)=6tan⁡x+10f ( x ) = 6 \tan x + 10f(x)=6tanx+10

Write the trigonometric expression as an algebraic expression in u. - cos⁡(tan⁡−1u)\cos \left( \tan ^ { - 1 } \mathrm { u } \right)cos(tan−1u)

Complete the identity. - tan⁡α+cot⁡β=?\tan \alpha + \cot \beta = ?tanα+cotβ=?

Establish the identity. - sin⁡3xcos⁡2x=sin⁡x(cos⁡2x−cos⁡4x)\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)sin3xcos2x=sinx(cos2x−cos4x)