Deck 29: Inverse Trigonometric Functions

Question 1

Use an inverse trigonometric function to write θ as a function of x.​ ​ a = x
B = 10
​

A) $\theta = \arcsin \frac { x } { 10 }$
B) $\theta = \arccos \frac { 10 } { x }$
C) $\theta = \operatorname { arccot } \frac { x } { 10 }$
D) $\theta = \arccos \frac { x } { 10 }$
E) $\theta = \arctan \frac { x } { 10 }$

Answer 1

$\theta = \arccos \frac { 10 } { x }$

Question 2

Use an inverse trigonometric function to write θ as a function of x.​ ​ a = x
B = 6
​

A) $\theta = \arctan \frac { 6 } { x }$
B) $\theta = \arccos \frac { x } { 6 }$
C) $\theta = \operatorname { arccot } \frac { x } { 6 }$
D) $\theta = \arcsin \frac { x } { 6 }$
E) $\theta = \arctan \frac { x } { 6 }$

Answer 2

$\theta = \arctan \frac { x } { 6 }$

Question 3

Evaluate the expression.Round your result to two decimal places.​ $$\cos ^ { - 1 } 0.44$$ ​

A)2.12
B)0.12
C)1.12
D)-0.88
E)3.12

Answer 3

1.12

Question 4

Use an inverse trigonometric function to write θ as a function of x.​ ​ a = x + 5
B = 10
​

A) $$\theta = \arccos \frac { x + 5 } { 10 }$$
B) $$\theta = \arctan \frac { x + 5 } { 10 }$$
C) $\theta = \arctan \frac { 10 } { x + 5 }$
D) $\theta = \operatorname { arccot } \frac { x + 5 } { 10 }$
E) $\theta = \arcsin \frac { x + 5 } { 10 }$

Question 5

Use an inverse trigonometric function to write θ as a function of x.​ ​ a = x + 6
B = 2
​

A) $\theta = \arctan \frac { x + 6 } { 2 }$
B) $\theta = \arcsin \frac { 2 } { x + 6 }$
C) $\theta = \operatorname { arccot } \frac { x + 6 } { 2 }$
D) $\theta = \arccos \frac { x + 6 } { 2 }$
E) $\theta = \arcsin \frac { x + 6 } { 2 }$

Question 6

Evaluate the expression.Round your result to two decimal places.​ $$\arcsin ( - 0.130 )$$ ​

A)1.87
B)-2.13
C)-1.13
D)0.87
E)-0.13

Question 7

Evaluate the expression.Round your result to two decimal places.​ $$\arctan 30$$ ​

A)3.54
B)0.54
C)1.54
D)2.54
E)-0.46

Question 8

Evaluate the expression.Round your result to two decimal places.​ $$\arccos 0.23$$ ​

A)-0.66
B)2.34
C)1.34
D)3.34
E)0.34

Question 9

Use the properties of inverse trigonometric functions to evaluate the expression.​ $$\tan ( \arctan 40 )$$ ​

A) $\tan ^ { - 1 } 40$
B) $- 40$
C) $\frac { \pi } { 40 }$
D) $\tan 40$
E)40

Question 10

Evaluate the expression.Round your result to two decimal places.​ $$\arccos \left( - \frac { 2 } { 3 } \right)$$ ​

A)0.30
B)1.30
C)4.30
D)3.30
E)2.30

Question 11

Evaluate the expression.Round your result to two decimal places.​ $$\arcsin 0.45$$ ​

A)-1.53
B)-0.53
C)2.47
D)0.47
E)1.47

Question 12

Evaluate the expression.Round your result to two decimal places.​ $$\tan ^ { - 1 } \left( - \frac { 97 } { 4 } \right)$$ ​

A)-3.53
B)0.47
C)-0.53
D)-1.53
E)-2.53

Question 13

Evaluate the expression.Round your result to two decimal places.​ $$\arccos ( - 0.6 )$$ ​

A)4.21
B)2.21
C)3.21
D)0.21
E)1.21

Question 14

Evaluate the expression.Round your result to two decimal places.​ $$\sin ^ { - 1 } 0.65$$ ​

A)2.71
B)-0.29
C)1.71
D)0.71
E)-1.29

Question 15

Evaluate the expression.Round your result to two decimal places.​ $$\tan ^ { - 1 } ( - \sqrt { 588 } )$$ ​

A)-0.53
B)-1.53
C)0.47
D)-3.53
E)-2.53

Question 16

Use the properties of inverse trigonometric functions to evaluate the expression.​ $\sin [ \arcsin ( - 0.4 ) ]$ ​

A) $$\sin ^ { - 1 } 0.4$$
B) $$\sin ( - 0.4 )$$
C)0.4
D) $- 0.4$
E) $\frac { \pi } { 0.4 }$

Question 17

Use an inverse trigonometric function to write θ as a function of x.​ ​ a = 5x
B = x + 9
​

A) $\theta = \arctan \frac { x + 9 } { 5 x }$
B) $\theta = \arcsin \frac { x + 9 } { 5 x }$
C) $\theta = \operatorname { arcsec } \frac { x + 9 } { 5 x }$
D) $\theta = \operatorname { arccsc } \frac { x + 9 } { 5 x }$
E) $\theta = \arccos \frac { x + 9 } { 5 x }$

Question 18

Use an inverse trigonometric function to write θ as a function of x.​ ​ $\begin{array} { l } a = x ^ { 2 } - 16 \\b = x - 4\end{array}$ ​

A) $\theta = \arcsin \frac { 1 } { x + 4 } , x \neq 4$
B) $\theta = \arctan \frac { 1 } { x - 4 } , x \neq 4$
C) $\theta = \arctan \frac { 1 } { x + 4 } , x \neq 4$
D) $\theta = \arccos \frac { 1 } { x + 4 } , x \neq 4$
E) $\theta = \operatorname { arccot } \frac { 1 } { x + 4 } , x \neq 4$

Question 19

Evaluate the expression.Round your result to two decimal places.​ $$\arcsin \frac { 2 } { 3 }$$ ​

A)1.73
B)-0.27
C)-1.27
D)2.73
E)0.73

Question 20

Evaluate the expression.Round your result to two decimal places.​ $$\arctan ~~2.6$$ ​

A)2.20
B)0.20
C)3.20
D)-0.80
E)1.20

Question 21

An airplane flies at an altitude of $a = 6$ miles toward a point directly over an observer.Consider θ and x as shown in the figure.Write θ as a function of x.​ ​

A) $\theta = \operatorname { arccsc } \frac { x } { 6 }$
B) $\theta = \operatorname { arccot } \frac { 6 } { x }$
C) $\theta = \arcsin \frac { x } { 6 }$
D) $\theta = \arctan \frac { 6 } { x }$
E) $\theta = \arctan \frac { x } { 6 }$

Question 22

Evaluate $\tan ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$ without using a calculator.

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

Question 23

Find an algebraic expression that is equivalent to the expression..​ $\cos ( \arcsin 4 x )$ ​

A) $\frac { \sqrt { 1 + 16 x ^ { 2 } } } { 1 + 16 x ^ { 2 } }$
B) $\frac { \sqrt { 1 - 16 x ^ { 2 } } } { 1 - 16 x ^ { 2 } }$
C) $\sqrt { 16 x ^ { 2 } - 1 }$
D) $\sqrt { 1 + 16 x ^ { 2 } }$
E) $\sqrt { 1 - 16 x ^ { 2 } }$

Question 24

Find an algebraic expression that is equivalent to the expression.​ $$\sec ( \arctan 4 x )$$ ​

A) $\sqrt { 16 x ^ { 2 } + 1 }$
B) $\sqrt { 1 - 16 x ^ { 2 } }$
C) $\frac { \sqrt { 16 x ^ { 2 } - 1 } } { 16 x ^ { 2 } - 1 }$
D) $\sqrt { 16 x ^ { 2 } - 1 }$
E) $\frac { \sqrt { 16 x ^ { 2 } + 1 } } { 16 x ^ { 2 } + 1 }$

Question 25

Find the value of the expression.Round your result to two decimal places.​ $$\operatorname { arccot } \frac { 7 } { 2 }$$ ​

A)0.28
B)1.28
C)1.29
D)-1.29
E)-0.28

Question 26

A television camera at ground level is filming the lift-off of a space shuttle at a point $a = 925$ meters from the launch pad (see figure).Let θ be the angle of elevation to the shuttle and let s be the height of the shuttle.Write θ as a function of s.​ ​

A) $\theta = \arctan \frac { s } { 925 }$
B) $\theta = \arcsin \frac { s } { 925 }$
C) $\theta = \operatorname { arccsc } \frac { s } { 925 }$
D) $\theta = \arctan \frac { 925 } { s }$
E) $\theta = \operatorname { arccot } \frac { s } { 925 }$

Question 27

A boat is pulled in by means of a winch located on a dock $a = 2$ feet above the deck of the boat (see figure).Let θ be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat.Write θ as a function of s.​ ​

A) $\theta = \operatorname { arccsc } \frac { 2 } { s }$
B) $\theta = \operatorname { arcsec } \frac { 2 } { s }$
C) $\theta = \arccos \frac { 2 } { s }$
D) $\theta = \arctan \frac { 2 } { s }$
E) $\theta = \arcsin \frac { 2 } { s }$

Question 28

Find the value of the expression.Round your result to two decimal places.​ $$\operatorname { arccsc } ( - 24 )$$ ​

A)1.61
B)1.53
C)0.04
D)0.96
E)-0.04

Question 29

Use the properties of inverse trigonometric functions to evaluate $$\tan [ \arctan ( - 0.11 ) ]$$ .

A) $- 0.55$
B) $0.12$
C) $- 0.41$
D) $- 0.11$
E) $- 0.07$

Question 30

A security car with its spotlight on is parked $a = 30$ meters from a warehouse.Consider θ and x as shown in the figure.Write θ as a function of x.​ ​

A) $\theta = \operatorname { arccot } \frac { x } { 30 }$
B) $\theta = \arctan \frac { x } { 30 }$
C) $\theta = \operatorname { arccsc } \frac { x } { 30 }$
D) $\theta = \arcsin \frac { x } { 30 }$
E) $\theta = \arctan \frac { 30 } { x }$

Question 31

Use a calculator to evaluate $\cos ^ { - 1 } ( - 0.81 )$ .Round your answer to two decimal places.

A)0.68
B) $- 0.89$
C)0.72
D) $- 0.94$
E)2.51

Question 32

Use the properties of inverse trigonometric functions to evaluate $$\arctan \left[ \tan \left( \frac { 2 \pi } { 5 } \right) \right]$$ .

A) $\frac { \pi } { 5 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 5 \pi } { 2 }$
D) $\frac { 2 \pi } { 5 }$
E) $- \frac { 3 \pi } { 5 }$

Question 33

Use an inverse function to write θ as a function of x.

A)​ $\theta = \tan ^ { - 1 } \left( \frac { 1 } { x - 1 } \right)$
B) $\theta = \tan ^ { - 1 } \left( \frac { x - 1 } { x + 1 } \right)$
C) $\theta = \tan ^ { - 1 } ( x + 1 )$
D) $\theta = \tan ^ { - 1 } \left( \frac { 1 } { x ^ { 2 } + 1 } \right)$
E) $\theta = \tan ^ { - 1 } \left( \frac { 1 } { x + 1 } \right)$

Question 34

Find the value of the expression.Round your result to two decimal places.​ $$\operatorname { arccsc } \left( - \frac { 23 } { 6 } \right)$$ ​

A)-1.32
B)1.83
C)-0.26
D)-1.26
E)0.74

Question 35

Use the properties of inverse trigonometric functions to evaluate the expression.​ $\arcsin ( \sin 7 \pi )$ ​

A)1
B)-1
C) $\cos ^ { - 1 } 7 \pi$
D)0
E) $\sin ^ { - 1 } 7 \pi$

Question 36

Use a calculator to evaluate $\arctan 0.95$ .Round your answer to two decimal places.

A)1.25
B)0.58
C)0.32
D)1.40
E)0.76

Question 37

Find an algebraic expression that is equivalent to the expression..​ $$\tan \left( \arccos \frac { x } { 6 } \right)$$ ​

A) $\frac { \sqrt { 36 - x ^ { 2 } } } { x }$
B) $\frac { 36 - x ^ { 2 } } { x }$
C) $\frac { \sqrt { 36 + x ^ { 2 } } } { x }$
D) $\frac { \sqrt { 36 + x ^ { 2 } } } { 36 + x ^ { 2 } }$
E) $\frac { \sqrt { 36 - x ^ { 2 } } } { 36 - x ^ { 2 } }$

Question 38

Evaluate $\arctan \frac { \sqrt { 3 } } { 3 }$ without using a calculator.

A) $\frac { \pi } { 6 }$
B) $- \frac { \pi } { 6 }$
C) $\frac { \pi } { 4 }$
D) $\frac { \pi } { 3 }$
E) $- \frac { 3 \pi } { 4 }$

Question 39

Find the value of the expression.Round your result to two decimal places.​ $\operatorname { arcsec }~~~~ 3.14$ ​

A)-1.25​
B)1.25
C)1.89
D)3.14
E)-3.14 ​

Question 40

Use an inverse function to write θ as a function of x.

A) $\theta = \arctan \frac { \sqrt { x ^ { 2 } - 9 } } { 3 }$
B) $\theta = \arctan \frac { 3 } { x }$
C) $\theta = \arctan \frac { 3 \pi } { x }$
D) $\theta = \arctan \frac { \sqrt { x ^ { 2 } + 9 } } { 3 }$
E) $\theta = \arctan \frac { x } { 3 }$

Question 41

Use a graphing utility to graph the function below.​ $y = 2 \arcsin ( 2 x )$ ​ π $- 1$ $1$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$ -π
Π $- 1$ $1$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$ -π

A)π​ $$\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$$ -π​
B)π​ $$\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$$ -π​
C)​ π​ $$\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$$ -π​
D)π​ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$ -π​
E)π​ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 4 }$ -π​

Question 42

Find the exact value of $$\tan \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right)$$ .

A) $\frac { 1 } { 5 }$
B) $\frac { 17 } { 12 }$
C) $\frac { 5 } { 13 }$
D) $\frac { 5 } { 12 }$
E) $\frac { 5 } { 17 }$

Question 43

Use a graphing utility to graph the function below. $y = 2 \sin ^ { - 1 } ( 2 x ) + \frac { \pi } { 2 }$

A) $$\frac { 3 \pi } { 2 }$$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 2 }$ $- \frac { \pi } { 2 }$
B) $\frac { 3 \pi } { 2 }$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 2 }$ $- \frac { \pi } { 2 }$
C) $\frac { 3 \pi } { 2 }$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 2 }$ $- \frac { \pi } { 2 }$
D) $\frac { 3 \pi } { 2 }$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 2 }$ $- \frac { \pi } { 2 }$
E) $\frac { 3 \pi } { 2 }$ $\mathrm { Y } _ { \mathrm { scl } } = \frac { \pi } { 2 }$ $- \frac { \pi } { 2 }$

Question 44

Write an algebraic expression that is equivalent to $$\sin \left( \arctan \frac { x } { 2 } \right)$$ .

A) $\frac { x \sqrt { x ^ { 2 } + 4 } } { x ^ { 2 } + 4 }$
B) $\frac { 2 \sqrt { x ^ { 2 } + 4 } } { x ^ { 2 } + 4 }$
C) $\frac { 2 } { x }$
D) $\frac { \sqrt { x ^ { 2 } + 4 } } { 2 }$
E) $\frac { \sqrt { x ^ { 2 } + 4 } } { x }$

Question 45

Write an algebraic expression that is equivalent to $\tan ( \arccos 3 x )$ .

A) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 3 x }$
B) $\sqrt { 1 - 9 x ^ { 2 } }$
C) $\frac { \sqrt { 1 - 9 x ^ { 2 } } } { 1 - 9 x ^ { 2 } }$
D) $\frac { 1 } { 3 x }$
E) $3 x$

Question 46

A granular substance such as sand naturally settles into a cone-shaped pile when poured from a small aperture.Its height depends on the humidity and adhesion between granules.The angle of elevation of a pile,θ,is called the angle of repose.If the height of a pile of sand is 14 feet and its diameter is approximately 47 feet,determine the angle of repose.Round your answer to the nearest degree.

A) $27 ^ { \circ }$
B) $29 ^ { \circ }$
C) $28 ^ { \circ }$
D) $30 ^ { \circ }$
E) $31 ^ { \circ }$

Question 47

Find the exact value of $$\sin \left( \arctan \frac { 7 } { 24 } \right)$$ .

A) $\frac { 24 } { 7 }$
B) $\frac { 7 } { 24 }$
C) $\frac { 7 } { 32 }$
D) $\frac { 7 } { 25 }$
E) $\frac { 32 } { 25 }$

Question 48

Which of the following functions is represented by the graph below?

A) $\arcsin \left( \frac { x - 1 } { 2 } \right) + \frac { \pi } { 4 }$
B) $2 \arccos ( x + 1 )$
C) $\arcsin ( 2 x )$
D) $\arccos \left( \frac { x + 1 } { 2 } \right)$
E) $2 \arcsin \frac { x } { 2 } - 1$

Question 49

Which of the following functions is represented by the graph below?

A)2arccos x
B)arcsin 2x
C)arccos (x + 1)
D)2arcsin $\frac { x } { 2 }$
E)arccos $\frac { x } { 2 }$

Question 50

Which of the following can be inserted to make the statement true? $$\arccos \left( \frac { \sqrt { 36 - x ^ { 2 } } } { 6 } \right) = \arcsin ( \quad ) , 0 \leq x \leq 6$$

A) $$\frac { x ^ { 2 } } { 6 }$$
B) $$\frac { x } { 6 }$$
C) $36 - x ^ { 2 }$
D) $\frac { \sqrt { 72 - x ^ { 2 } } } { x }$
E)x