Question 1

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

Accepted Answer

The answer of Prove the identity.
-\[- \tan ^ { 2...

Question 2

Choose the one alternative that best completes the statement or answers the question.
-Under which of the following conditions do we know that two triangles are congruent? (More than one may app
(i) Three sides of one triangle are equal to the corresponding sides of the second triangle.
(ii) Three angles of one triangle are equal to the corresponding angles of the second triangle.
(iii) Two angles and the included side of one triangle are equal to the corresponding parts of the second triangle.

Accepted Answer

A)  (iii) only 
B)  (i), (ii), (iii) 
C)  (i) only 
D)  (i), (iii) 
A)  (iii) only 
B)  (i), (ii), (iii) 
C)  (i) only 
D)  (i), (iii)

Question 3

State whether the given measurements determine zero, one, or two triangles.
-$$\mathrm { A } = 53 ^ { \circ } , \mathrm { a } = 21 , \mathrm {~b} = 22$$

Accepted Answer

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

Question 4

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

Accepted Answer

A)  $( \sin x + 2 ) ( \sin x - 1 )$ 
B)  $( \sin x - 2 ) ( \sin x - 1 )$ 
C)  $( \sin x - 2 ) ( \sin x + 1 )$ 
D)  $( \sin x + 2 ) ( \sin x + 1 )$ 
A)  $( \sin x + 2 ) ( \sin x - 1 )$ 
B)  $( \sin x - 2 ) ( \sin x - 1 )$ 
C)  $( \sin x - 2 ) ( \sin x + 1 )$ 
D)  $( \sin x + 2 ) ( \sin x + 1 )$

Question 5

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

Accepted Answer

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

Question 6

Solve the triangle.
-$$\mathrm { B } = 36 ^ { \circ } , \mathrm { a } = 39 , \mathrm { c } = 19$$

Accepted Answer

A)  $b \approx 26.1 , C \approx 115.9 , \mathrm {~A} \approx 28.1$ 
B)  $b \approx 43 , C \approx 29.1 , \mathrm {~A} \approx 114.9$ 
C)  $b \approx 26.1 , C \approx 25.1 , A \approx 118.9$ 
D)  $b \approx 43 , C \approx 25.1 , A \approx 118.9$ 
A)  $b \approx 26.1 , C \approx 115.9 , \mathrm {~A} \approx 28.1$ 
B)  $b \approx 43 , C \approx 29.1 , \mathrm {~A} \approx 114.9$ 
C)  $b \approx 26.1 , C \approx 25.1 , A \approx 118.9$ 
D)  $b \approx 43 , C \approx 25.1 , A \approx 118.9$

Question 7

Prove 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 8

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

Accepted Answer

A)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C)  $0 , \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  No solution 
A)  $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$ 
B)  $\frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
C)  $0 , \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  No solution

Question 9

Solve.
-A boat leaves the dock and sails in a direction of $70 ^ { \circ }$. Once reaching its destination on the opposite shore, it sails in a direction of $272 ^ { \circ }$ and docks $150 \mathrm {~km}$ north of its original starting position. What is the total distance the boat has traveled?

Accepted Answer

A)  $997 \mathrm {~km}$ 
B)  $776 \mathrm {~km}$ 
C)  $926 \mathrm {~km}$ 
D)  $626 \mathrm {~km}$ 
A)  $997 \mathrm {~km}$ 
B)  $776 \mathrm {~km}$ 
C)  $926 \mathrm {~km}$ 
D)  $626 \mathrm {~km}$

Question 10

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the
triangle.
-a = 14 b = 8
C = 4

Accepted Answer

A)  24.19 
B)  25.10 
C)  No triangle is formed. 
D)  23.24 
A)  24.19 
B)  25.10 
C)  No triangle is formed. 
D)  23.24

Question 11

Find all solutions in the interval [0, 2π]
-$7 \tan ^ { 3 } x - 21 \tan x = 0$

Accepted Answer

A)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $0 , \frac { \pi } { 5 } , \pi , \frac { 6 \pi } { 5 }$ 
D)  $0 , \frac { \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
A)  $\frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
C)  $0 , \frac { \pi } { 5 } , \pi , \frac { 6 \pi } { 5 }$ 
D)  $0 , \frac { \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$

Question 12

Prove the identity.
-$$\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$$

Accepted Answer

The answer of Prove the identity.
-\[\frac { \cos t }...

Question 13

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

Accepted Answer

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

Question 14

Express the function as a sinusoid of the form y = a sin (bx + c).
-$$y = 3 \sin ( 4 x ) + 4 \cos ( 4 x )$$

Accepted Answer

A)  $y = 5 \sin \left( 4 x - \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ ) 
B)  $y = 5 \sin \left( 4 x - \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ 
C)  $y = 5 \sin \left( 4 x + \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ 
D)  $y = 5 \sin \left( 4 x + \sin - 1 \left( \frac { 4 } { 5 } \right) \right)$ 
A)  $y = 5 \sin \left( 4 x - \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ ) 
B)  $y = 5 \sin \left( 4 x - \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ 
C)  $y = 5 \sin \left( 4 x + \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right)$ 
D)  $y = 5 \sin \left( 4 x + \sin - 1 \left( \frac { 4 } { 5 } \right) \right)$

Question 15

Write the expression as the sine, cosine, or tangent of an angle.
-$$\sin \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \cos \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$$

Accepted Answer

A)  $\sin \frac { - 4 \pi } { 77 }$ 
B)  $\cos \frac { 18 \pi } { 77 }$ 
C)  $\sin \frac { 18 \pi } { 77 }$ 
D)  $\cos \frac { - 4 \pi } { 77 }$ 
A)  $\sin \frac { - 4 \pi } { 77 }$ 
B)  $\cos \frac { 18 \pi } { 77 }$ 
C)  $\sin \frac { 18 \pi } { 77 }$ 
D)  $\cos \frac { - 4 \pi } { 77 }$

Question 16

Complete the identity.
-The expression $\frac { 1 + \tan ^ { 2 } x } { \tan ^ { 2 } x }$
is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be u: the right hand side of the equation to complete the identity?

Accepted Answer

A)  $- \cos ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\csc ^ { 2 } x$ 
D)  $\tan ^ { 2 } x$ 
A)  $- \cos ^ { 2 } x$ 
B)  $\sec ^ { 2 } x$ 
C)  $\csc ^ { 2 } x$ 
D)  $\tan ^ { 2 } x$

Question 17

Find the exact value by using a half-angle identity.
-$\sin \frac { 7 \pi } { 8 }$

Accepted Answer

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

Question 18

Rewrite with only sin x and cos x.
-$\cos 2 x + \sin x$

Accepted Answer

A)  $1 - 2 \sin ^ { 2 } x + \sin x$ 
B)  $1 + 2 \sin ^ { 2 } x + \sin x$ 
C)  $1 + 3 \sin ^ { 2 } x$ 
D)  $1 + 3 \sin x$ 
A)  $1 - 2 \sin ^ { 2 } x + \sin x$ 
B)  $1 + 2 \sin ^ { 2 } x + \sin x$ 
C)  $1 + 3 \sin ^ { 2 } x$ 
D)  $1 + 3 \sin x$

Question 19

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 6 }$ and $\sin \theta < 0$

Accepted Answer

A)  $- \sqrt { 37 }$ 
B)  $- \frac { \sqrt { 35 } } { 35 }$ 
C)  $- \sqrt { 35 }$ 
D)  6 
A)  $- \sqrt { 37 }$ 
B)  $- \frac { \sqrt { 35 } } { 35 }$ 
C)  $- \sqrt { 35 }$ 
D)  6

Question 20

Rewrite with only sin x and cos x.
-$\sin 2 x - \cos 3 x$

Accepted Answer

A)  $2 \sin ^ { 2 } x \cos x - \cos ^ { 3 } x - 2 \sin x \cos x$ 
B)  $3 \sin ^ { 2 } x \cos x - \sin ^ { 3 } x + 2 \sin x \cos x$ 
C)  $2 \sin x \cos x + \cos x - 4 \cos x \sin ^ { 2 } x$ 
D)  $\cos ^ { 3 } x + 2 \sin ^ { 2 } x \cos x - \sin ^ { 2 } x + 2 \sin x \cos x$ 
A)  $2 \sin ^ { 2 } x \cos x - \cos ^ { 3 } x - 2 \sin x \cos x$ 
B)  $3 \sin ^ { 2 } x \cos x - \sin ^ { 3 } x + 2 \sin x \cos x$ 
C)  $2 \sin x \cos x + \cos x - 4 \cos x \sin ^ { 2 } x$ 
D)  $\cos ^ { 3 } x + 2 \sin ^ { 2 } x \cos x - \sin ^ { 2 } x + 2 \sin x \cos x$

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

State whether the given measurements determine zero, one, or two triangles. - $\mathrm { A } = 53 ^ { \circ } , \mathrm { a } = 21 , \mathrm {~b} = 22$

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

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

Solve the triangle. - $\mathrm { B } = 36 ^ { \circ } , \mathrm { a } = 39 , \mathrm { c } = 19$

Prove the identity. - $\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }$

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

Solve. -A boat leaves the dock and sails in a direction of $70 ^ { \circ }$ . Once reaching its destination on the opposite shore, it sails in a direction of $272 ^ { \circ }$ and docks $150 \mathrm {~km}$ north of its original starting position. What is the total distance the boat has traveled?

Decide whether a triangle can be formed with the given side lengths. If so, use Heron's formula to find the area of the triangle. -a = 14 b = 8 C = 4

Find all solutions in the interval [0, 2π] - $7 \tan ^ { 3 } x - 21 \tan x = 0$

Prove the identity. - $\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$

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

Express the function as a sinusoid of the form y = a sin (bx + c). - $y = 3 \sin ( 4 x ) + 4 \cos ( 4 x )$

Write the expression as the sine, cosine, or tangent of an angle. - $\sin \frac { \pi } { 7 } \cos \frac { \pi } { 11 } + \cos \frac { \pi } { 7 } \sin \frac { \pi } { 11 }$

Complete the identity. -The expression $\frac { 1 + \tan ^ { 2 } x } { \tan ^ { 2 } x }$ is to be the left hand side of an equation that is an identity. Which one of the following four expressions can be u: the right hand side of the equation to complete the identity?

Find the exact value by using a half-angle identity. - $\sin \frac { 7 \pi } { 8 }$

Rewrite with only sin x and cos x. - $\cos 2 x + \sin x$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 6 }$ and $\sin \theta < 0$

Rewrite with only sin x and cos x. - $\sin 2 x - \cos 3 x$

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Exam 5: Analytic Trigonometry

Prove the identity. - −tan⁡2x+sec⁡2x=sec⁡2xcos⁡2x- \tan ^ { 2 } x + \sec ^ { 2 } x = \sec ^ { 2 } x \cos ^ { 2 } x−tan2x+sec2x=sec2xcos2x

State whether the given measurements determine zero, one, or two triangles. - A=53∘,a=21, b=22\mathrm { A } = 53 ^ { \circ } , \mathrm { a } = 21 , \mathrm {~b} = 22A=53∘,a=21, b=22

Write each expression in factored form as an algebraic expression of a single trigonometric function. - sin⁡2x+sin⁡x−2\sin ^ { 2 } x + \sin x - 2sin2x+sinx−2

Find all solutions to the equation in the interval [0, 2). - 2cos⁡x+sin⁡2x=02 \cos x + \sin 2 x = 02cosx+sin2x=0

Solve the triangle. - B=36∘,a=39,c=19\mathrm { B } = 36 ^ { \circ } , \mathrm { a } = 39 , \mathrm { c } = 19B=36∘,a=39,c=19

Prove the identity. - cos⁡(x−y)cos⁡(x+y)=1+tan⁡xtan⁡y1−tan⁡xtan⁡y\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }cos(x+y)cos(x−y)​=1−tanxtany1+tanxtany​

Find all solutions to the equation in the interval [0, 2). - cos⁡2x−cos⁡x=0\cos 2 x - \cos x = 0cos2x−cosx=0