Question 1

Which of the following is equivalent to the given expression? 
$\frac { \cos ^ { 2 } x } { 1 + \sin x }$

Accepted Answer

A)   $\tan x + \cos x$ 
B)   $1 - \sin x$ 
C)   $\csc x + \cot x$ 
D)   $\cot x - \sin x$ 
E)   $\cot x \cos x + \tan x$ 
A)   $\tan x + \cos x$ 
B)   $1 - \sin x$ 
C)   $\csc x + \cot x$ 
D)   $\cot x - \sin x$ 
E)   $\cot x \cos x + \tan x$

Question 2

Find the exact solutions of the given equation in the interval [0, 2$\pi$).   $\cos 2 x + 3 \cos x + 2 = 0$

Accepted Answer

A)   $x = 0 , \pi , \frac { 2 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)   $x = \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)   $x = \pi , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
D)  x = 0 
E)   $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
A)   $x = 0 , \pi , \frac { 2 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)   $x = \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)   $x = \pi , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
D)  x = 0 
E)   $x = 0 , \pi , \frac { \pi } { 3 } , \frac { 4 \pi } { 3 }$

Question 3

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Accepted Answer

A)  $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
B)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
A)  $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
B)  $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
C)  $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$ 
D)  $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 4

Solve the following equation.   $9 \cot ^ { 2 } x - 3 = 0$

Accepted Answer

A)   $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$ 
B)   $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$ 
C)  $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$ 
D)   $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$ 
E)  $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$ 
A)   $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$ 
B)   $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$ 
C)  $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$ 
D)   $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$ 
E)  $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$

Question 5

A Ferris wheel is built such that the height h (in feet) above the ground of a seat on the wheel at time t (in seconds) can be modeled by $h ( t ) = 67 + 54 \sin \left( \frac { \pi } { 18 } t - \frac { \pi } { 2 } \right)$ .The wheel makes one revolution every 36 seconds and the ride begins when t = 0.During the first 36 seconds of the ride, when will a person, who starts at the bottom of the Ferris wheel, be 67 feet above the ground?

Accepted Answer

A) 9 seconds and 22 seconds 
B) 9 seconds and 27 seconds 
C) 10 seconds and 22 seconds 
D) 9 seconds and 17 seconds 
E) 10 seconds and 17 seconds 
A) 9 seconds and 22 seconds 
B) 9 seconds and 27 seconds 
C) 10 seconds and 22 seconds 
D) 9 seconds and 17 seconds 
E) 10 seconds and 17 seconds

Question 6

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta $, where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 36 - 9 x ^ { 2 } } , x = 2 \cos \theta$

Accepted Answer

A) 2 cos $\theta $ 
B) 36 sin $\theta $ 
C) - 2 sin$\theta $ 
D) 36 cos $\theta $ 
E) 6 sin $\theta $ 
A) 2 cos $\theta $ 
B) 36 sin $\theta $ 
C) - 2 sin$\theta $ 
D) 36 cos $\theta $ 
E) 6 sin $\theta $

Question 7

Use the cofunction identities to evaluate the expression without using a calculator. ​
Tan273° + cot214° - sec276° - csc217°
​

Accepted Answer

A) 3 
B) 1 
C) -1 
D) -2 
E) 2 
A) 3 
B) 1 
C) -1 
D) -2 
E) 2

Question 8

The mach number M of an airplane is the ratio of its speed to the speed of sound.When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure).The mach number is related to the apex angle $\theta $ of the cone by $\sin ( \theta / 5 ) = 1 / M$ .      
Rewrite the equation in terms of $\theta$.

Accepted Answer

A)   $\theta = 5 \sin \left( \frac { 1 } { M } \right)$ 
B)   $\theta = \sin \left( \frac { 1 } { M } \right)$ 
C)   $\theta = \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$ 
D)   $\theta = 5 \sin ^ { - 1 } \left( \frac { 5 } { M } \right)$ 
E)  $\theta = 5 \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$ 
A)   $\theta = 5 \sin \left( \frac { 1 } { M } \right)$ 
B)   $\theta = \sin \left( \frac { 1 } { M } \right)$ 
C)   $\theta = \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$ 
D)   $\theta = 5 \sin ^ { - 1 } \left( \frac { 5 } { M } \right)$ 
E)  $\theta = 5 \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$

Question 9

The monthly sales S (in hundreds of units) of baseball equipment for an Internet sporting goods site are approximated by $S = 55.7 - 37.5 \cos \frac { \pi t } { 6 }$ where t is the time (in months), with t = 1 corresponding to January.Determine the months when sales exceed 7700 units at any time during the month.

Accepted Answer

A) April through August 
B) July through May 
C) June through August 
D) April through July 
E) May through July 
A) April through August 
B) July through May 
C) June through August 
D) April through July 
E) May through July

Question 10

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) ​
2 cos x + 2 sin x tan x
​

Accepted Answer

A) -2 sec x 
B) 2 sec x 
C) 4 cos x 
D) -4 cos x 
E) 3 sin x 
A) -2 sec x 
B) 2 sec x 
C) 4 cos x 
D) -4 cos x 
E) 3 sin x

Question 11

Find all solutions of the following equation in the interval [0, 2$\pi $). 
 $5 \sec ^ { 2 } x - 5 = 0$

Accepted Answer

A)  $0 , \frac { \pi } { 2 }$ 
B)   $\pi , \frac { 2 \pi } { 3 }$ 
C)   $0 , \pi$ 
D)   $0 , \frac { 5 \pi } { 3 }$ 
E)   $2 \pi , \frac { 3 \pi } { 4 }$ 
A)  $0 , \frac { \pi } { 2 }$ 
B)   $\pi , \frac { 2 \pi } { 3 }$ 
C)   $0 , \pi$ 
D)   $0 , \frac { 5 \pi } { 3 }$ 
E)   $2 \pi , \frac { 3 \pi } { 4 }$

Question 12

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta $, where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .Then find sin $\theta $and cos $\theta $. 
$3 \sqrt { 3 } = \sqrt { 81 - 9 x ^ { 2 } } , x = 3 \cos \theta$

Accepted Answer

A)   $\sin \theta = 3 ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
B)   $3 \sin \theta = 3 ; \sin \theta = 0 ; \cos \theta = 1$ 
C)  $\sin \theta = \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
D)   $9 \sin \theta = 3 \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
E)   $3 \cos \theta = 3 ; \sin \theta = 1 ; \cos \theta = 0$ 
A)   $\sin \theta = 3 ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
B)   $3 \sin \theta = 3 ; \sin \theta = 0 ; \cos \theta = 1$ 
C)  $\sin \theta = \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
D)   $9 \sin \theta = 3 \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$ 
E)   $3 \cos \theta = 3 ; \sin \theta = 1 ; \cos \theta = 0$

Question 13

Simplify the expression algebraically. 
 $\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$

Accepted Answer

A)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$ 
B)  $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$ 
C)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$ 
D)  $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$ 
E)  $\cos ^ { 2 } x - \sin ^ { 2 } y$ 
A)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$ 
B)  $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$ 
C)  $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$ 
D)  $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$ 
E)  $\cos ^ { 2 } x - \sin ^ { 2 } y$

Question 14

Simplify the following expression algebraically. 
 $6 \tan ( \pi + \theta )$

Accepted Answer

A)  $- \frac { 3 } { 2 } \tan \theta$ 
B)  $- 6 \sin \theta$ 
C)  $- 6 \tan \theta$ 
D)  $6 \tan \theta$ 
E)  $\frac { 3 } { 2 } \tan \theta$ 
A)  $- \frac { 3 } { 2 } \tan \theta$ 
B)  $- 6 \sin \theta$ 
C)  $- 6 \tan \theta$ 
D)  $6 \tan \theta$ 
E)  $\frac { 3 } { 2 } \tan \theta$

Question 15

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta $, where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta$

Accepted Answer

A) 25 tan$\theta $ 
B) 5 tan $\theta $ 
C) 25 sec$\theta $ 
D) 6 sec $\theta $ 
E) - 5 sec $\theta $ 
A) 25 tan$\theta $ 
B) 5 tan $\theta $ 
C) 25 sec$\theta $ 
D) 6 sec $\theta $ 
E) - 5 sec $\theta $

Question 16

Use the sum-to-product formulas to write the given expression as a product.  
  $\sin 5 \theta - \sin 3 \theta$

Accepted Answer

A)  2 sin 4$\theta$cos $\theta$ 
B)  -2 sin 4$\theta$  sin$\theta$ 
C) - 2 cos 4$\theta$  cos$\theta$ 
D)  2 cos 4$\theta$ sin $\theta$ 
E)  2 cos 4$\theta$ cos $\theta$ 
A)  2 sin 4$\theta$cos $\theta$ 
B)  -2 sin 4$\theta$  sin$\theta$ 
C) - 2 cos 4$\theta$  cos$\theta$ 
D)  2 cos 4$\theta$ sin $\theta$ 
E)  2 cos 4$\theta$ cos $\theta$

Question 17

A weight is attached to a spring suspended vertically from a ceiling.When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by 
 $y = \frac { 1 } { 8 } \sin 2 t + \frac { 1 } { 6 } \cos 2 t$ where y is the distance from equilibrium (in feet) and t is the time (in seconds).
Find the amplitude of the oscillations of the weight.

Accepted Answer

A)  $\frac { 1 } { 24 } \mathrm { ft }$ 
B)  $\frac { 1 } { 10 } \mathrm { ft }$ 
C)  $\frac { 5 } { 24 } \mathrm { ft }$ 
D)  $\frac { 1 } { 5 } \mathrm { ft }$ 
E)  $\frac { 24 } { 5 } \mathrm { ft }$ 
A)  $\frac { 1 } { 24 } \mathrm { ft }$ 
B)  $\frac { 1 } { 10 } \mathrm { ft }$ 
C)  $\frac { 5 } { 24 } \mathrm { ft }$ 
D)  $\frac { 1 } { 5 } \mathrm { ft }$ 
E)  $\frac { 24 } { 5 } \mathrm { ft }$

Question 18

Solve the following equation.  
Secx - 2 = 0

Accepted Answer

A)   $x = \frac { \pi } { 5 } + 2 n \pi$ and $x = \frac { 7 \pi } { 5 } + 2 n \pi$ , where n is a integer 
B)   $x = \frac { \pi } { 3 } + 2 n \pi$ and $x = \frac { 5 \pi } { 3 } + 2 n \pi$ , where n is a integer 
C)   $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 7 \pi } { 6 } + 2 n \pi$ , where n is a integer 
D)   $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 5 \pi } { 6 } + 2 n \pi$ , where n is a integer 
E)   $x = \frac { \pi } { 4 } + 2 n \pi$ and $x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is a integer 
A)   $x = \frac { \pi } { 5 } + 2 n \pi$ and $x = \frac { 7 \pi } { 5 } + 2 n \pi$ , where n is a integer 
B)   $x = \frac { \pi } { 3 } + 2 n \pi$ and $x = \frac { 5 \pi } { 3 } + 2 n \pi$ , where n is a integer 
C)   $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 7 \pi } { 6 } + 2 n \pi$ , where n is a integer 
D)   $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 5 \pi } { 6 } + 2 n \pi$ , where n is a integer 
E)   $x = \frac { \pi } { 4 } + 2 n \pi$ and $x = \frac { 5 \pi } { 4 } + 2 n \pi$ , where n is a integer

Question 19

Use the sum-to-product formulas to rewrite the sum or difference as a product.   $\sin 9 \theta + \sin 7 \theta$

Accepted Answer

A)   $2 \sin 8 \theta \sin \theta$ 
B)   $2 \cos 8 \theta \cos \theta$ 
C)   $2 \sin 8 \theta \cos \theta$ 
D)   $2 \cos 8 \theta \sin \theta$ 
E)   $2 \sin 9 \theta \cos 7 \theta$ 
A)   $2 \sin 8 \theta \sin \theta$ 
B)   $2 \cos 8 \theta \cos \theta$ 
C)   $2 \sin 8 \theta \cos \theta$ 
D)   $2 \cos 8 \theta \sin \theta$ 
E)   $2 \sin 9 \theta \cos 7 \theta$

Question 20

Find the rate of change of the function $f ( x ) = - x + \cos x$ .

Accepted Answer

A)   $- \sin x + 1$ 
B)   $\sin x - 1$ 
C)   $- \sin x - 1$ 
D)   $- \sin x - x$ 
E)   $\sin x + 1$ 
A)   $- \sin x + 1$ 
B)   $\sin x - 1$ 
C)   $- \sin x - 1$ 
D)   $- \sin x - x$ 
E)   $\sin x + 1$

Which of the following is equivalent to the given expression? $\frac { \cos ^ { 2 } x } { 1 + \sin x }$

Find the exact solutions of the given equation in the interval [0, 2 $\pi$ ). $\cos 2 x + 3 \cos x + 2 = 0$

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Solve the following equation. $9 \cot ^ { 2 } x - 3 = 0$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 36 - 9 x ^ { 2 } } , x = 2 \cos \theta$

Use the cofunction identities to evaluate the expression without using a calculator. Tan²73° + cot²14° - sec²76° - csc²17°

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) 2 cos x + 2 sin x tan x

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $5 \sec ^ { 2 } x - 5 = 0$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .Then find sin $\theta$ and cos $\theta$ . $3 \sqrt { 3 } = \sqrt { 81 - 9 x ^ { 2 } } , x = 3 \cos \theta$

Simplify the expression algebraically. $\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$

Simplify the following expression algebraically. $6 \tan ( \pi + \theta )$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta$

Use the sum-to-product formulas to write the given expression as a product. $\sin 5 \theta - \sin 3 \theta$

Solve the following equation. Secx - 2 = 0

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 9 \theta + \sin 7 \theta$

Find the rate of change of the function $f ( x ) = - x + \cos x$ .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 5: Analytic Trigonometry

Which of the following is equivalent to the given expression? cos⁡2x1+sin⁡x\frac { \cos ^ { 2 } x } { 1 + \sin x }1+sinxcos2x​

Find the exact solutions of the given equation in the interval [0, 2 π\piπ ). cos⁡2x+3cos⁡x+2=0\cos 2 x + 3 \cos x + 2 = 0cos2x+3cosx+2=0

Find the exact value of the given expression using a sum or difference formula. sin⁡285∘\sin 285 ^ { \circ }sin285∘

Solve the following equation. 9cot⁡2x−3=09 \cot ^ { 2 } x - 3 = 09cot2x−3=0

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ\theta θ , where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ . 36−9x2,x=2cos⁡θ\sqrt { 36 - 9 x ^ { 2 } } , x = 2 \cos \theta36−9x2​,x=2cosθ

Use the cofunction identities to evaluate the expression without using a calculator. ​ Tan273° + cot214° - sec276° - csc217° ​

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) ​ 2 cos x + 2 sin x tan x ​

Find all solutions of the following equation in the interval [0, 2 π\pi π ). 5sec⁡2x−5=05 \sec ^ { 2 } x - 5 = 05sec2x−5=0

Simplify the expression algebraically. cos⁡(7x+4y)cos⁡(7x−4y)\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )cos(7x+4y)cos(7x−4y)

Simplify the following expression algebraically. 6tan⁡(π+θ)6 \tan ( \pi + \theta )6tan(π+θ)

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ\theta θ , where 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ . 25x2+36,5x=6tan⁡θ\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta25x2+36​,5x=6tanθ

Use the sum-to-product formulas to write the given expression as a product. sin⁡5θ−sin⁡3θ\sin 5 \theta - \sin 3 \thetasin5θ−sin3θ

Solve the following equation. Secx - 2 = 0

Use the sum-to-product formulas to rewrite the sum or difference as a product. sin⁡9θ+sin⁡7θ\sin 9 \theta + \sin 7 \thetasin9θ+sin7θ

Find the rate of change of the function f(x)=−x+cos⁡xf ( x ) = - x + \cos xf(x)=−x+cosx .

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Which of the following is equivalent to the given expression? $\frac { \cos ^ { 2 } x } { 1 + \sin x }$

Find the exact solutions of the given equation in the interval [0, 2 $\pi$ ). $\cos 2 x + 3 \cos x + 2 = 0$

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

Solve the following equation. $9 \cot ^ { 2 } x - 3 = 0$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 36 - 9 x ^ { 2 } } , x = 2 \cos \theta$

Use the cofunction identities to evaluate the expression without using a calculator. Tan²73° + cot²14° - sec²76° - csc²17°

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) 2 cos x + 2 sin x tan x

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $5 \sec ^ { 2 } x - 5 = 0$

Simplify the expression algebraically. $\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$

Simplify the following expression algebraically. $6 \tan ( \pi + \theta )$

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of $\theta$ , where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta$

Use the sum-to-product formulas to write the given expression as a product. $\sin 5 \theta - \sin 3 \theta$

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 9 \theta + \sin 7 \theta$

Find the rate of change of the function $f ( x ) = - x + \cos x$ .