Question 1

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

Accepted Answer

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)  $\frac { \sqrt { x ^ { 2 } + 4 } } { x }$ 
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 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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)  $\frac { \sqrt { 16 x ^ { 2 } + 1 } } { 16 x ^ { 2 } + 1 }$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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)  $\theta = \arctan \frac { x } { 10 }$ 
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)  $\theta = \arctan \frac { x } { 10 }$

Question 11

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.​   ​

Accepted Answer

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)  $\theta = \arcsin \frac { 2 } { s }$ 
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 12

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

Accepted Answer

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)  $\theta = \arcsin \frac { x + 5 } { 10 }$ 
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 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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)  $2 \arcsin \frac { x } { 2 } - 1$ 
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 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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)  $\sqrt { 1 - 16 x ^ { 2 } }$ 
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 20

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

Accepted Answer

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

Exam 29: Inverse Trigonometric Functions

Write an algebraic expression that is equivalent to sin⁡(arctan⁡x2)\sin \left( \arctan \frac { x } { 2 } \right)sin(arctan2x​) .

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

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

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

Find an algebraic expression that is equivalent to the expression.​ sec⁡(arctan⁡4x)\sec ( \arctan 4 x )sec(arctan4x) ​

Use a calculator to evaluate arctan⁡0.95\arctan 0.95arctan0.95 .Round your answer to two decimal places.

Find the exact value of tan⁡(sin⁡−1513)\tan \left( \sin ^ { - 1 } \frac { 5 } { 13 } \right)tan(sin−1135​) .

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

Find the exact value of sin⁡(arctan⁡724)\sin \left( \arctan \frac { 7 } { 24 } \right)sin(arctan247​) .

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

A boat is pulled in by means of a winch located on a dock a=2a = 2a=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.​ ​

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

Find the value of the expression.Round your result to two decimal places.​ arccsc⁡(−236)\operatorname { arccsc } \left( - \frac { 23 } { 6 } \right)arccsc(−623​) ​

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

Use the properties of inverse trigonometric functions to evaluate the expression.​ tan⁡(arctan⁡40)\tan ( \arctan 40 )tan(arctan40) ​

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

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

Use the properties of inverse trigonometric functions to evaluate the expression.​ arcsin⁡(sin⁡7π)\arcsin ( \sin 7 \pi )arcsin(sin7π) ​

Find an algebraic expression that is equivalent to the expression..​ cos⁡(arcsin⁡4x)\cos ( \arcsin 4 x )cos(arcsin4x) ​

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

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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