Question 1

Find the domain of the function f and of its inverse function f-1.
-$$f ( x ) = 3 \sin ( 5 x )$$

Accepted Answer

A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 2,8 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5,5 ]$ 
C)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 3,3 ]$ 
D)  Domain of $\mathrm { f } : \left[ - \frac { 1 } { 5 } , \frac { 1 } { 5 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 2,8 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5,5 ]$ 
C)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 3,3 ]$ 
D)  Domain of $\mathrm { f } : \left[ - \frac { 1 } { 5 } , \frac { 1 } { 5 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Question 2

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

Accepted Answer

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

Question 3

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Write the word or phrase that best completes each statement or answers the question.
-If two sound sources at the same volume are equidistant from a microphone, the pressure on the microphone is $\varepsilon$ by
$$p = a \cos \omega _ { 1 } t + a \cos \omega _ { 2 } t$$
where $a _ { , } \omega _ { 1 } , \omega _ { 2 }$ are constants and $t$ is time. Write $\mathrm { p }$ as a product of cosine functions.

Accepted Answer

The answer of Write the word or phrase that best...

Question 5

Find the inverse function f-1 of the function f.
-$$f ( x ) = 5 \tan ( 10 x - 9 )$$

Accepted Answer

A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 5 \tan ^ { - 1 } ( 10 \mathrm { x } - 9 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 5 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) + 9 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) - 9 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) + 9 \right]$ 
A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 5 \tan ^ { - 1 } ( 10 \mathrm { x } - 9 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 5 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) + 9 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) - 9 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) + 9 \right]$

Question 6

Find the exact value of the expression.
-$\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 13 } { 4 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 13 } { 4 }$ 
D)  $- \frac { 1 } { 2 }$

Question 7

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\sin 22.5 ^ { \circ }$

Accepted Answer

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

Question 8

Choose the one alternative that best completes the statement or answers the question.
Simplify the expression.
-$$\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta$$

Accepted Answer

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

Question 9

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\sin 75 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
A)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
B)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question 10

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$$\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$$

Accepted Answer

A)  $2 \sin \theta \cos \theta$ 
B)  $\cos ( 2 \theta ) + \sin \theta \cos \theta$ 
C)  $2 \cos ( 2 \theta )$ 
D)  $4 \cos ^ { 2 } \theta - 1$ 
A)  $2 \sin \theta \cos \theta$ 
B)  $\cos ( 2 \theta ) + \sin \theta \cos \theta$ 
C)  $2 \cos ( 2 \theta )$ 
D)  $4 \cos ^ { 2 } \theta - 1$

Question 11

Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 3 } { 5 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \cos \beta = - \frac { \sqrt { 21 } } { 5 } , \pi < \beta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( \alpha - \beta )$

Accepted Answer

A)  $\frac { - 6 - 4 \sqrt { 21 } } { 25 }$ 
B)  $\frac { - 8 + 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$ 
A)  $\frac { - 6 - 4 \sqrt { 21 } } { 25 }$ 
B)  $\frac { - 8 + 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$

Question 12

Find the exact value under the given conditions.
-$\cos \alpha = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = \frac { 15 } { 17 } , \frac { \pi } { 2 } < \alpha < \pi \quad$ Find $\tan ( \alpha + \beta )$

Accepted Answer

A)  $- \frac { 171 } { 221 }$ 
B)  $\frac { 171 } { 140 }$ 
C)  $- \frac { 171 } { 220 }$ 
D)  $\frac { 3 } { 20 }$ 
A)  $- \frac { 171 } { 221 }$ 
B)  $\frac { 171 } { 140 }$ 
C)  $- \frac { 171 } { 220 }$ 
D)  $\frac { 3 } { 20 }$

Question 13

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

Accepted Answer

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

Question 14

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 15

Write the word or phrase that best completes each statement or answers the question.
-The path of a projectile fired at an inclination θ (in degrees) to the horizontal with an initial speed v0 is a
parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the
formula $$\mathrm { R } = \frac { \mathrm { v } \mathrm { v } _ { 0 } ^ { 2 } } { \mathrm {~g} } \sin ( 2 \theta$$ where g is the acceleration due to gravity. The maximum height H of the projectile is $$\mathrm { H } = \frac { \mathrm { v } _ { 0 } ^ { 2 } } { 4 \mathrm {~g} } ( 1 - \cos ( 2 \theta ) )$$ Find the range R and the maximum height H in terms of g if the projectile is fired with an initial speed of 200 meters per
second at an angle of 15° and then at an angle of 22.5°. Do not use a calculator, but simplify the answers.

Accepted Answer

\[\begin{array} { l } 
\theta = 15 ^ { \

Question 16

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }$$

Accepted Answer

\[\begin{array} { l } 
\frac { \cos ( x

Question 17

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

Accepted Answer

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

Question 18

Solve the problem.
-If $\sin \theta = - \frac { 4 } { 5 }$, and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$, then find $\tan 2 \theta$.

Accepted Answer

A)  $\frac { 7 } { 24 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $- \frac { 7 } { 24 }$ 
D)  $\frac { 24 } { 7 }$ 
A)  $\frac { 7 } { 24 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $- \frac { 7 } { 24 }$ 
D)  $\frac { 24 } { 7 }$

Question 19

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } x$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 20

Write the word or phrase that best completes each statement or answers the question.
-$$\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$$

Accepted Answer

\[\begin{array} { l } 
\cos 3 x = \cos (

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = 3 \sin ( 5 x )$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$

Find the inverse function f-1 of the function f. - $f ( x ) = 5 \tan ( 10 x - 9 )$

Find the exact value of the expression. - $\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 22.5 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Simplify the expression. - $\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 75 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$

Find the exact value under the given conditions. - $\sin \alpha = - \frac { 3 } { 5 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \cos \beta = - \frac { \sqrt { 21 } } { 5 } , \pi < \beta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( \alpha - \beta )$

Find the exact value under the given conditions. - $\cos \alpha = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = \frac { 15 } { 17 } , \frac { \pi } { 2 } < \alpha < \pi \quad$ Find $\tan ( \alpha + \beta )$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }$

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

Solve the problem. -If $\sin \theta = - \frac { 4 } { 5 }$ , and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$ , then find $\tan 2 \theta$ .

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } x$

Write the word or phrase that best completes each statement or answers the question. - $\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 6: Analytic Trigonometry

Find the domain of the function f and of its inverse function f-1. - f(x)=3sin⁡(5x)f ( x ) = 3 \sin ( 5 x )f(x)=3sin(5x)

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - cot⁡3x=cot⁡x(csc⁡2x−1)\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)cot3x=cotx(csc2x−1)

Find the inverse function f-1 of the function f. - f(x)=5tan⁡(10x−9)f ( x ) = 5 \tan ( 10 x - 9 )f(x)=5tan(10x−9)

Find the exact value of the expression. - sin⁡195∘cos⁡75∘−cos⁡195∘sin⁡75∘\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }sin195∘cos75∘−cos195∘sin75∘

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡22.5∘\sin 22.5 ^ { \circ }sin22.5∘

Choose the one alternative that best completes the statement or answers the question. Simplify the expression. - cos⁡θ1+sin⁡θ+tan⁡θ\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta1+sinθcosθ​+tanθ

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡75∘\sin 75 ^ { \circ }sin75∘

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - sin⁡(3θ)sin⁡θ=?\frac { \sin ( 3 \theta ) } { \sin \theta } = ?sinθsin(3θ)​=?

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - cos⁡t1+sin⁡t+1+sin⁡tcos⁡t=2sec⁡t\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t1+sintcost​+cost1+sint​=2sect

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

Solve the problem. -If sin⁡θ=−45\sin \theta = - \frac { 4 } { 5 }sinθ=−54​ , and 3π2<θ<2π\frac { 3 \pi } { 2 } < \theta < 2 \pi23π​<θ<2π , then find tan⁡2θ\tan 2 \thetatan2θ .

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - sec⁡4x−tan⁡4x=sec⁡2x+tan⁡2x\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } xsec4x−tan4x=sec2x+tan2x

Write the word or phrase that best completes each statement or answers the question. - cos⁡3x=cos⁡3x−3sin⁡2xcos⁡x\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos xcos3x=cos3x−3sin2xcosx

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = 3 \sin ( 5 x )$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cot ^ { 3 } x = \cot x \left( \csc ^ { 2 } x - 1 \right)$

Find the inverse function f-1 of the function f. - $f ( x ) = 5 \tan ( 10 x - 9 )$

Find the exact value of the expression. - $\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 22.5 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Simplify the expression. - $\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 75 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$

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

Solve the problem. -If $\sin \theta = - \frac { 4 } { 5 }$ , and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$ , then find $\tan 2 \theta$ .

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } x$

Write the word or phrase that best completes each statement or answers the question. - $\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$