Question 1

State whether the given measurements determine zero, one, or two triangles.
-$$B = 71 ^ { \circ } , b = 19 , c = 14$$

Accepted Answer

A)  Two 
B)  One 
C)  Zero 
A)  Two 
B)  One 
C)  Zero

Question 2

Simplify the expression.
-$$\left( \sin ^ { 2 } x + \cos ^ { 2 } x \right) - \left( \csc ^ { 2 } x - \cot ^ { 2 } x \right)$$

Accepted Answer

A)  $\sin ^ { 2 } x$ 
B)  $\cos ^ { 2 } x$ 
C)  0 
D)  2 
A)  $\sin ^ { 2 } x$ 
B)  $\cos ^ { 2 } x$ 
C)  0 
D)  2

Question 3

Use basic identities to simplify the expression.
-$\sin \theta \cos \theta \sec \theta \csc \theta$

Accepted Answer

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

Question 4

Find all solutions in the interval [0, 2π]
-$$\tan x + \sec x = 1$$

Accepted Answer

A)  $x = \frac { 5 \pi } { 4 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = 0$ 
D)  No solution 
A)  $x = \frac { 5 \pi } { 4 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = 0$ 
D)  No solution

Question 5

Use basic identities to simplify the expression.
-$$\frac { \tan \theta } { \cot \theta }$$

Accepted Answer

A)  $\sin \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$ 
A)  $\sin \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\cos ^ { 3 } \theta$

Question 6

Find the exact value by using a half-angle identity.
-$\sin \frac { \pi } { 12 }$

Accepted Answer

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

Question 7

Write the expression as the sine, cosine, or tangent of an angle.
-$\cos 7 x \cos 4 x - \sin 7 x \sin 4 x$

Accepted Answer

A)  $\sin 11 x$ 
B)  $\sin 3 x$ 
C)  $\cos 11 \mathrm { x }$ 
D)  $\cos 3 x$ 
A)  $\sin 11 x$ 
B)  $\sin 3 x$ 
C)  $\cos 11 \mathrm { x }$ 
D)  $\cos 3 x$

Question 8

Find all solutions to the equation.
-$\sin ( \cos x ) = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 2 } + 2 n \pi , n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } + \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
D)  $\{ \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$ 
A)  $\left\{ \frac { \pi } { 2 } + 2 n \pi , n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
B)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \mid n = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
C)  $\left\{ \frac { \pi } { 2 } + \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \right\}$ 
D)  $\{ \mathrm { n } \pi \mid \mathrm { n } = 0 , \pm 1 , \pm 2 , \ldots . \}$

Question 9

Find all solutions in the interval [0, 2π]
-$$\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$$

Accepted Answer

A)  $x = \frac { \pi } { 3 }$ 
B)  No solution 
C)  $x = \frac { \pi } { 6 }$ 
D)  $x = \frac { \pi } { 4 }$ 
A)  $x = \frac { \pi } { 3 }$ 
B)  No solution 
C)  $x = \frac { \pi } { 6 }$ 
D)  $x = \frac { \pi } { 4 }$

Question 10

Find an exact value.
-$$\tan \frac { 13 \pi } { 12 }$$

Accepted Answer

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

Question 11

Prove the identity.
-$$\sec \left( \frac { \pi } { 2 } + u \right) = - \csc u$$

Accepted Answer

The answer of Prove the identity.
-\[\sec \left( \frac { \pi...

Question 12

Find all solutions to the equation in the interval [0, 2).
-$\cos x - \cos 3 x = 0$

Accepted Answer

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

Question 13

Write the expression as the sine, cosine, or tangent of an angle.
-$\frac { \tan 5 x - \tan 4 y } { 1 + \tan 5 x \tan 4 y }$

Accepted Answer

A)  $\tan ( 5 x - 4 y )$ 
B)  $\tan \left( \frac { 5 x - 4 y } { 2 } \right)$ 
C)  $\tan \left( \frac { 5 x + 4 y } { 2 } \right)$ 
D)  $\tan ( 5 x + 4 y )$ 
A)  $\tan ( 5 x - 4 y )$ 
B)  $\tan \left( \frac { 5 x - 4 y } { 2 } \right)$ 
C)  $\tan \left( \frac { 5 x + 4 y } { 2 } \right)$ 
D)  $\tan ( 5 x + 4 y )$

Question 14

Find all solutions in the interval [0, 2π]
-$$\sin ^ { 2 } x - \cos ^ { 2 } x = 0$$

Accepted Answer

A)  $x = \frac { \pi } { 4 } , \frac { \pi } { 3 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $x = \frac { \pi } { 4 } , \frac { \pi } { 6 }$ 
A)  $x = \frac { \pi } { 4 } , \frac { \pi } { 3 }$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $x = \frac { \pi } { 4 } , \frac { \pi } { 6 }$

Question 15

Prove the identity.
-$$\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$$

Accepted Answer

The answer of Prove the identity.
-\[\cot x \sec ^ {...

Question 16

Find an exact value.
-$\sin 75 ^ { \circ }$

Accepted Answer

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

Question 17

Write each expression in factored form as an algebraic expression of a single trigonometric function.
-$$\frac { \sin ^ { 2 } x - 1 } { 1 + \sin x }$$

Accepted Answer

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

Question 18

Find all solutions in the interval [0, 2π]
-$4 \sin ^ { 2 } x - 4 \sin x + 1 = 0$

Accepted Answer

A)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
A)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
B)  $\frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $\frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$

Question 19

Use basic identities to simplify the expression.
-$\cot \theta \sec \theta \sin \theta$

Accepted Answer

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

Question 20

Find an exact value.
-$\tan 105 ^ { \circ }$

Accepted Answer

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

State whether the given measurements determine zero, one, or two triangles. - $B = 71 ^ { \circ } , b = 19 , c = 14$

Simplify the expression. - $\left( \sin ^ { 2 } x + \cos ^ { 2 } x \right) - \left( \csc ^ { 2 } x - \cot ^ { 2 } x \right)$

Use basic identities to simplify the expression. - $\sin \theta \cos \theta \sec \theta \csc \theta$

Find all solutions in the interval [0, 2π] - $\tan x + \sec x = 1$

Use basic identities to simplify the expression. - $\frac { \tan \theta } { \cot \theta }$

Find the exact value by using a half-angle identity. - $\sin \frac { \pi } { 12 }$

Write the expression as the sine, cosine, or tangent of an angle. - $\cos 7 x \cos 4 x - \sin 7 x \sin 4 x$

Find all solutions to the equation. - $\sin ( \cos x ) = 0$

Find all solutions in the interval [0, 2π] - $\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$

Find an exact value. - $\tan \frac { 13 \pi } { 12 }$

Prove the identity. - $\sec \left( \frac { \pi } { 2 } + u \right) = - \csc u$

Find all solutions to the equation in the interval [0, 2). - $\cos x - \cos 3 x = 0$

Write the expression as the sine, cosine, or tangent of an angle. - $\frac { \tan 5 x - \tan 4 y } { 1 + \tan 5 x \tan 4 y }$

Find all solutions in the interval [0, 2π] - $\sin ^ { 2 } x - \cos ^ { 2 } x = 0$

Prove the identity. - $\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$

Find an exact value. - $\sin 75 ^ { \circ }$

Write each expression in factored form as an algebraic expression of a single trigonometric function. - $\frac { \sin ^ { 2 } x - 1 } { 1 + \sin x }$

Find all solutions in the interval [0, 2π] - $4 \sin ^ { 2 } x - 4 \sin x + 1 = 0$

Use basic identities to simplify the expression. - $\cot \theta \sec \theta \sin \theta$

Find an exact value. - $\tan 105 ^ { \circ }$

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An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 5: Analytic Trigonometry

State whether the given measurements determine zero, one, or two triangles. - B=71∘,b=19,c=14B = 71 ^ { \circ } , b = 19 , c = 14B=71∘,b=19,c=14

Simplify the expression. - (sin⁡2x+cos⁡2x)−(csc⁡2x−cot⁡2x)\left( \sin ^ { 2 } x + \cos ^ { 2 } x \right) - \left( \csc ^ { 2 } x - \cot ^ { 2 } x \right)(sin2x+cos2x)−(csc2x−cot2x)

Use basic identities to simplify the expression. - sin⁡θcos⁡θsec⁡θcsc⁡θ\sin \theta \cos \theta \sec \theta \csc \thetasinθcosθsecθcscθ

Find all solutions in the interval [0, 2π] - tan⁡x+sec⁡x=1\tan x + \sec x = 1tanx+secx=1

Use basic identities to simplify the expression. - tan⁡θcot⁡θ\frac { \tan \theta } { \cot \theta }cotθtanθ​

Find the exact value by using a half-angle identity. - sin⁡π12\sin \frac { \pi } { 12 }sin12π​

Write the expression as the sine, cosine, or tangent of an angle. - cos⁡7xcos⁡4x−sin⁡7xsin⁡4x\cos 7 x \cos 4 x - \sin 7 x \sin 4 xcos7xcos4x−sin7xsin4x

Find all solutions to the equation. - sin⁡(cos⁡x)=0\sin ( \cos x ) = 0sin(cosx)=0

Find all solutions in the interval [0, 2π] - sec⁡2x−2=tan⁡2x\sec ^ { 2 } x - 2 = \tan ^ { 2 } xsec2x−2=tan2x

Find an exact value. - tan⁡13π12\tan \frac { 13 \pi } { 12 }tan1213π​

Prove the identity. - sec⁡(π2+u)=−csc⁡u\sec \left( \frac { \pi } { 2 } + u \right) = - \csc usec(2π​+u)=−cscu

Find all solutions to the equation in the interval [0, 2). - cos⁡x−cos⁡3x=0\cos x - \cos 3 x = 0cosx−cos3x=0

Write the expression as the sine, cosine, or tangent of an angle. - tan⁡5x−tan⁡4y1+tan⁡5xtan⁡4y\frac { \tan 5 x - \tan 4 y } { 1 + \tan 5 x \tan 4 y }1+tan5xtan4ytan5x−tan4y​

Find all solutions in the interval [0, 2π] - sin⁡2x−cos⁡2x=0\sin ^ { 2 } x - \cos ^ { 2 } x = 0sin2x−cos2x=0

Prove the identity. - cot⁡xsec⁡4x=cot⁡x+2tan⁡x+tan⁡3x\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } xcotxsec4x=cotx+2tanx+tan3x

Find an exact value. - sin⁡75∘\sin 75 ^ { \circ }sin75∘

Write each expression in factored form as an algebraic expression of a single trigonometric function. - sin⁡2x−11+sin⁡x\frac { \sin ^ { 2 } x - 1 } { 1 + \sin x }1+sinxsin2x−1​

Find all solutions in the interval [0, 2π] - 4sin⁡2x−4sin⁡x+1=04 \sin ^ { 2 } x - 4 \sin x + 1 = 04sin2x−4sinx+1=0

Use basic identities to simplify the expression. - cot⁡θsec⁡θsin⁡θ\cot \theta \sec \theta \sin \thetacotθsecθsinθ

Find an exact value. - tan⁡105∘\tan 105 ^ { \circ }tan105∘