Question 1

Solve.
-Consider the formula T = 591 - 76 cos 2ϴ, where ϴ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of ϴ for which the value of T will be 577?

Accepted Answer

A) 79.38° 
B) 39.69° 
C) 82.38° 
D) 36.69°

Question 2

Solve the equation for exact solutions.
-$\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }$

Accepted Answer

A)  $\left\{ \frac { 3 - \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { 3 + \pi } { 3 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { 3 + \pi } { 6 } \right\}$

Question 3

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-Explain what is wrong with the following solution for the equation $\sin 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi$ ).
$$\sin 2 \theta = \sqrt { 3 }$$
$$\sin \theta = \frac { \sqrt { 3 } } { 2 }$$
$$\theta = \frac { \pi } { 3 } \text { or } \theta = \frac { 2 \pi } { 3 }$$

Accepted Answer

The answer of Use the parallelogram rule to find the...

Question 4

Solve the equation for exact solutions.
-$\arctan x = \arcsin \frac { 3 } { 5 }$

Accepted Answer

A)  $\left\{ \frac { 5 } { 4 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ \frac { 4 } { 3 } \right\}$ 
D)  $\left\{ \frac { 3 } { 4 } \right\}$

Question 5

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$2 \sqrt { 3 } \sin 4 x = 3$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$

Question 6

Solve the equation for x, where x is restricted to the given interval.
-$$y = - \sec \frac { x } { 4 } \text {, for } x \text { in } [ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]$$

Accepted Answer

A)  $x = \frac { 1 } { 4 } \operatorname { arcsec } ( - y )$ 
B)  $x = 4 \operatorname { arcsec } ( - y )$ 
C)  $x = - \frac { 1 } { 4 } \operatorname { arcsec } y$ 
D)  $x = - 4 \operatorname { arcsec } y$

Question 7

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$$\cot \frac { \theta } { 3 } = 1$$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 135 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$

Question 8

33 Solve the equation for solutions over the interval $[ 0,2 \pi )$. Write solutions as exact values or to four decimal places, as appropriate.
-$\tan 2 x + \sec 2 x = 2$

Accepted Answer

A)  $\{ 2.2143,8.4975 \}$ 
B)  $\{ 0.6435,6.9267 \}$ 
C)  $\{ 0.3218,3.4634 \}$ 
D)  $\{ 1.1072,4.2488 \}$

Question 9

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-On the given graph of $y = \csc x$ sketch the graph of $y = \csc ^ { - 1 } x$ as defined in the text. Give the domain and the range.

Accepted Answer

The answer of Use the parallelogram rule to find the...

Question 10

Provide an appropriate response.
-$$\text { True or false? The statement } \cos ^ { - 1 } ( \cos x ) = x \text { for all real numbers in the interval } \rightarrow \infty \leq x \leq \infty \text {. }$$

Accepted Answer

A) True 
 B)False

Question 11

Use a calculator to give the value to the nearest degree.
-$$\theta = \cos ^ { - 1 } ( 0.8910 )$$

Accepted Answer

A)  $63 ^ { \circ }$ 
B)  $25 ^ { \circ }$ 
C)  $27 ^ { \circ }$ 
D)  $29 ^ { \circ }$

Question 12

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\cot ( - \arcsin \pi )$

Accepted Answer

A)  $- 1094.771$ 
B)  0 
C)  Undefined 
D)  $- 0.0009$

Question 13

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi ) .$$
-$\sin ^ { 2 } 2 x = 1$

Accepted Answer

A)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } \right\}$ 
B)  $\varnothing$ 
C)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$

Question 14

Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers
in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$$\cos ^ { 2 } x - 1 = 0$$

Accepted Answer

A)  $\{ \mathrm { n } \pi \}$ 
B)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \right\}$ 
C)  $\{ 2 n \pi \}$ 
D)  $\left\{ \frac { \pi } { 3 } + n \pi \right\}$

Question 15

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$$\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$$

Accepted Answer

A)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$

Question 16

Give the degree measure of .
-$$\theta = \csc ^ { - 1 } ( 2 )$$

Accepted Answer

A)  $30 ^ { \circ }$ 
B)  $60 ^ { \circ }$ 
C)  $- 30 ^ { \circ }$ 
D)  $420 ^ { \circ }$

Question 17

Solve the problem.
-It can be shown that if the angle of elevation from an observer to the top of an object is A and the angle of elevation d ft closer is B, then the height of the object is given by $$\mathrm { h } = \frac { \mathrm { d } } { \cot \mathrm { A } - \cot \mathrm { B } } \mathrm { ft } \text {. }$$
Find $\mathrm { A }$ if $\mathrm { h } = 50 \mathrm { ft } , \mathrm { d } = 20 \mathrm { ft }$, and $\mathrm { B } = 61 ^ { \circ }$. Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $55.72 ^ { \circ }$ 
B)  $39.97 ^ { \circ }$ 
C)  $46.34 ^ { \circ }$ 
D)  $51.51 ^ { \circ }$

Question 18

Give the exact value of the expression.
-$\csc \left( \sin ^ { - 1 } \frac { 12 } { 20 } \right)$

Accepted Answer

A)  $\frac { 12 } { 20 }$ 
B)  $\frac { 12 } { 16 }$ 
C)  $\frac { 16 } { 12 }$ 
D)  $\frac { 20 } { 12 }$

Question 19

Give the degree measure of .
-$\theta = \cot ^ { - 1 } ( - \sqrt { 3 } )$

Accepted Answer

A)  $210 ^ { \circ }$ 
B)  $300 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $150 ^ { \circ }$

Question 20

Give the degree measure of .
-$$\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$$

Accepted Answer

A)  $120 ^ { \circ }$ 
B)  $45 ^ { \circ }$ 
C)  $315 ^ { \circ }$ 
D)  $135 ^ { \circ }$

Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve. -Consider the formula T = 591 - 76 cos 2ϴ, where ϴ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of ϴ for which the value of T will be 577?

Solve the equation for exact solutions. - $\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }$

Solve the equation for exact solutions. - $\arctan x = \arcsin \frac { 3 } { 5 }$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $2 \sqrt { 3 } \sin 4 x = 3$

Solve the equation for x, where x is restricted to the given interval. - $y = - \sec \frac { x } { 4 } \text {, for } x \text { in } [ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]$

Solve the equation for solutions in the interval [0°, 360°) . Round to the nearest degree. - $\cot \frac { \theta } { 3 } = 1$

33 Solve the equation for solutions over the interval $[ 0,2 \pi )$ . Write solutions as exact values or to four decimal places, as appropriate. - $\tan 2 x + \sec 2 x = 2$

Provide an appropriate response. - $\text { True or false? The statement } \cos ^ { - 1 } ( \cos x ) = x \text { for all real numbers in the interval } \rightarrow \infty \leq x \leq \infty \text {. }$

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( 0.8910 )$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\cot ( - \arcsin \pi )$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) .$ - $\sin ^ { 2 } 2 x = 1$

Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $\cos ^ { 2 } x - 1 = 0$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$

Give the degree measure of . - $\theta = \csc ^ { - 1 } ( 2 )$

Give the exact value of the expression. - $\csc \left( \sin ^ { - 1 } \frac { 12 } { 20 } \right)$

Give the degree measure of . - $\theta = \cot ^ { - 1 } ( - \sqrt { 3 } )$

Give the degree measure of . - $\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$

Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve. -Consider the formula T = 591 - 76 cos 2ϴ, where ϴ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of ϴ for which the value of T will be 577?

Solve the equation for exact solutions. - arcsin⁡(y−π6) =π6\arcsin \left( y - \frac { \pi } { 6 } \right) = \frac { \pi } { 6 }arcsin(y−6π​) =6π​

Solve the equation for exact solutions. - arctan⁡x=arcsin⁡35\arctan x = \arcsin \frac { 3 } { 5 }arctanx=arcsin53​

Solve the equation for solutions in the interval [0, 2 [0,2π) [ 0,2 \pi ) [0,2π) - 23sin⁡4x=32 \sqrt { 3 } \sin 4 x = 323​sin4x=3

Solve the equation for x, where x is restricted to the given interval. - y=−sec⁡x4, for x in [0,2π) ∪(2π,4π]y = - \sec \frac { x } { 4 } \text {, for } x \text { in } [ 0,2 \pi ) \cup ( 2 \pi , 4 \pi ]y=−sec4x​, for x in [0,2π) ∪(2π,4π]

Solve the equation for solutions in the interval [0°, 360°) . Round to the nearest degree. - cot⁡θ3=1\cot \frac { \theta } { 3 } = 1cot3θ​=1

33 Solve the equation for solutions over the interval [0,2π) [ 0,2 \pi ) [0,2π) . Write solutions as exact values or to four decimal places, as appropriate. - tan⁡2x+sec⁡2x=2\tan 2 x + \sec 2 x = 2tan2x+sec2x=2

Use a calculator to give the value to the nearest degree. - θ=cos⁡−1(0.8910) \theta = \cos ^ { - 1 } ( 0.8910 ) θ=cos−1(0.8910)

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - cot⁡(−arcsin⁡π) \cot ( - \arcsin \pi ) cot(−arcsinπ)

Solve the equation for solutions in the interval [0, 2 [0,2π) .[ 0,2 \pi ) .[0,2π) . - sin⁡22x=1\sin ^ { 2 } 2 x = 1sin22x=1

Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - cos⁡2x−1=0\cos ^ { 2 } x - 1 = 0cos2x−1=0

Solve the equation for solutions in the interval [0, 2 [0,2π) [ 0,2 \pi ) [0,2π) - sec⁡x2=cos⁡x2\sec \frac { x } { 2 } = \cos \frac { x } { 2 }sec2x​=cos2x​

Give the degree measure of . - θ=csc⁡−1(2) \theta = \csc ^ { - 1 } ( 2 ) θ=csc−1(2)

Give the exact value of the expression. - csc⁡(sin⁡−11220) \csc \left( \sin ^ { - 1 } \frac { 12 } { 20 } \right) csc(sin−12012​)

Give the degree measure of . - θ=cot⁡−1(−3) \theta = \cot ^ { - 1 } ( - \sqrt { 3 } ) θ=cot−1(−3​)

Give the degree measure of . - θ=sec⁡−1(−2) \theta = \sec ^ { - 1 } ( - \sqrt { 2 } ) θ=sec−1(−2​)