Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

3. Unit Circle

Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Trigonometric Functions on the Unit Circle

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2:31 minutes

Problem 28

Textbook Question

Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2.cos(θ + 20°)

Verified step by step guidance

Identify the cofunction identity for cosine: \( \cos(\theta) = \sin(90^\circ - \theta) \).

Apply the cofunction identity to the expression \( \cos(\theta + 20^\circ) \).

Rewrite \( \cos(\theta + 20^\circ) \) using the cofunction identity: \( \cos(\theta + 20^\circ) = \sin(90^\circ - (\theta + 20^\circ)) \).

Simplify the expression inside the sine function: \( 90^\circ - (\theta + 20^\circ) = 90^\circ - \theta - 20^\circ \).

The expression becomes \( \sin(70^\circ - \theta) \).

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Key Concepts

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

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles. For acute angles, the sine of an angle is equal to the cosine of its complement, and vice versa. This means sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). Understanding these identities is crucial for rewriting functions in terms of their cofunctions.

Angle Addition Formulas

The angle addition formulas allow us to express the sine and cosine of the sum of two angles in terms of the sines and cosines of the individual angles. For example, cos(θ + φ) = cos(θ)cos(φ) - sin(θ)sin(φ). These formulas are essential for simplifying expressions involving sums of angles, such as cos(θ + 20°).

Acute Angles

Acute angles are angles that measure less than 90 degrees. In trigonometry, the properties and values of trigonometric functions are often defined specifically for acute angles, as they yield positive values for sine and cosine. Recognizing that the angles involved are acute helps in applying the appropriate identities and formulas without concern for negative values or undefined expressions.