Table of contents

0. Review of College Algebra4h 45m

Rationalizing Denominators
17m

Pythagorean Theorem & Basics of Triangles
17m

Basics of Graphing
30m

Functions
41m

Transformations
45m

Asymptotes
4m

Solving Linear Equations
31m

Solving Quadratic Equations
55m

Complex Numbers
41m

1. Measuring Angles40m

Angles in Standard Position
12m

Coterminal Angles
8m

Complementary and Supplementary Angles
10m

Radians
8m

2. Trigonometric Functions on Right Triangles2h 5m

Trigonometric Functions on Right Triangles
45m

Special Right Triangles
30m

Cofunctions of Complementary Angles
26m

Solving Right Triangles
23m

3. Unit Circle1h 19m

Defining the Unit Circle
14m

Trigonometric Functions on the Unit Circle
9m

Common Values of Sine, Cosine, & Tangent
11m

Reference Angles
38m

Reciprocal Trigonometric Functions on the Unit Circle
6m

4. Graphing Trigonometric Functions1h 19m

Graphs of the Sine and Cosine Functions
32m

Phase Shifts
14m

Graphs of Secant and Cosecant Functions
10m

Graphs of Tangent and Cotangent Functions
21m

5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m

Inverse Sine, Cosine, & Tangent
28m

Evaluate Composite Trig Functions
44m

Linear Trigonometric Equations
28m

6. Trigonometric Identities and More Equations2h 34m

Introduction to Trigonometric Identities
1h 4m

Sum and Difference Identities
1h 3m

Double Angle Identities
16m

Solving Trigonometric Equations Using Identities
9m

7. Non-Right Triangles1h 38m

Law of Sines
49m

Law of Cosines
30m

Area of SAS & ASA Triangles
19m

8. Vectors2h 25m

Geometric Vectors
28m

Vectors in Component Form
32m

Direction of a Vector
23m

Unit Vectors and i & j Notation
27m

Dot Product
22m

Cross Product
11m

9. Polar Equations2h 5m

Polar Coordinate System
29m

Convert Points Between Polar and Rectangular Coordinates
26m

Convert Equations Between Polar and Rectangular Forms
29m

Graphing Other Common Polar Equations
39m

10. Parametric Equations1h 6m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

Writing Parametric Equations
21m

11. Graphing Complex Numbers1h 7m

Graphing Complex Numbers
6m

Polar Form of Complex Numbers
22m

Products and Quotients of Complex Numbers
14m

Powers of Complex Numbers (DeMoivre's Theorem)
23m

3. Unit Circle

Defining the Unit Circle

Trigonometry

3. Unit Circle

Defining the Unit Circle

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2:37 minutes

Problem 79

Textbook Question

Express each angular speed in radians per second. 6 revolutions per second

Verified step by step guidance

Recall that one revolution corresponds to an angle of $2\pi$ radians.

To convert revolutions per second to radians per second, multiply the number of revolutions per second by $2\pi$ radians per revolution.

Set up the conversion: angular speed in radians per second = (6 revolutions/second) $\times$ $2\pi$ radians/revolution.

Multiply the numerical values: $6 \times 2\pi$ to express the angular speed in terms of $\pi$.

Write the final expression for angular speed in radians per second as $12\pi$ radians per second (without calculating the decimal value).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Speed

Angular speed measures how fast an object rotates or revolves, typically expressed in radians per second. It quantifies the angle covered per unit time, helping to describe rotational motion precisely.

Radians and Revolutions

A radian is the standard unit of angular measure, defined as the angle subtended by an arc equal in length to the radius of the circle. One full revolution equals 2π radians, linking revolutions to radians for conversion.

Unit Conversion from Revolutions to Radians

To convert angular speed from revolutions per second to radians per second, multiply the number of revolutions by 2π. This conversion is essential for expressing angular velocity in standard SI units.

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Related Practice

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.tan (-11𝜋/6)

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Textbook Question

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋.4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

cos 9𝜋/2

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Textbook Question

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

tan 𝜋

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Multiple Choice

On the unit circle, what is the length of the radius?

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Multiple Choice

Find a polar equation for the curve represented by the given Cartesian equation $x^{2} + y^{2} = 2$ .

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Textbook Question