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Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Angles

For the following exercises, convert the angle measures to degrees.

1.

π 4     π 4    

2.

5π 3 5π 3

For the following exercises, convert the angle measures to radians.

3.

−210° −210°

4.

180° 180°

5.

Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85°. 85°.

6.

Find the area of the sector of a circle with diameter 32 feet and an angle of 3π 5 3π 5 radians.

For the following exercises, find the angle between and 360° 360° that is coterminal with the given angle.

7.

420° 420°

8.

−80° −80°

For the following exercises, find the angle between 0 and 2π 2π in radians that is coterminal with the given angle.

9.

20π 11 20π 11

10.

14π 5 14π 5

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

11.

−210° −210°

12.

75° 75°

13.

5π 4 5π 4

14.

π 3 π 3

15.

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour. Round to the nearest hundredth.

16.

A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour? Round to the nearest hundredth.

Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.

17.

cos π 4 cos π 4

18.

cot π 3 cot π 3

19.

tan π 6 tan π 6

20.

cos( π 2 )=sin( ___° ) cos( π 2 )=sin( ___° )

21.

csc( 18° )=sec( ___° ) csc( 18° )=sec( ___° )

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

22.

cosB= 3 5 ,a=6 cosB= 3 5 ,a=6

23.

tanA= 5 9 ,b=6 tanA= 5 9 ,b=6

For the following exercises, use Figure 1 to evaluate each trigonometric function.

A right triangle with side lengths of 11 and 6. Corners A and B are also labeled.  The angle A is opposite the side labeled 11.  The angle B is opposite the side labeled 6.
Figure 1
24.

sin A sin A

25.

tanB tanB

For the following exercises, solve for the unknown sides of the given triangle.

26.


A right triangle with corners labeled A, B, and C. Hypotenuse has length of 4 times square root of 2. Other angles measure 45 degrees.
27.


A right triangle with hypotenuse with length 5, and an angle of 30 degrees.
28.

A 15-ft ladder leans against a building so that the angle between the ground and the ladder is 70°. 70°. How high does the ladder reach up the side of the building? Find the answer to four decimal places.

29.

The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Find the answer to four decimal places.

Unit Circle

30.

Find the exact value of sin π 3 . sin π 3 .

31.

Find the exact value of cos π 4 . cos π 4 .

32.

Find the exact value of cosπ. cosπ.

33.

State the reference angle for 300°. 300°.

34.

State the reference angle for 3π 4 . 3π 4 .

35.

Compute cosine of 330°. 330°.

36.

Compute sine of 5π 4 . 5π 4 .

37.

State the domain of the sine and cosine functions.

38.

State the range of the sine and cosine functions.

The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.

39.

cos π 6 cos π 6

40.

tan π 4 tan π 4

41.

csc π 3 csc π 3

42.

sec π 4 sec π 4

For the following exercises, use reference angles to evaluate the given expression.

43.

sec 11π 3 sec 11π 3

44.

sec315° sec315°

45.

If sec( t )=−2.5, sec( t )=−2.5, what is the sec(t)? sec(t)?

46.

If tan(t)=−0.6, tan(t)=−0.6, what is the tan(t)? tan(t)?

47.

If tan(t)= 1 3 , tan(t)= 1 3 , find tan(tπ). tan(tπ).

48.

If cos(t)= 2 2 , cos(t)= 2 2 , find sin(t+2π). sin(t+2π). There are two possible solutions.

49.

Which trigonometric functions are even?

50.

Which trigonometric functions are odd?

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