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

6. Trigonometric Identities and More Equations

Sum and Difference Identities

Trigonometry

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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Problem 5.36

Textbook Question

Use identities to fill in each blank with the appropriate trigonometric function name.
tan 24° = 1/ _____ 66°

Verified step by step guidance

Recognize that the problem involves complementary angles, where the sum of the angles is 90°.

Recall the complementary angle identity: \( \tan(\theta) = \cot(90° - \theta) \).

Identify that \( \theta = 24° \) and \( 90° - \theta = 66° \).

Substitute \( \theta = 24° \) into the identity: \( \tan(24°) = \cot(66°) \).

Conclude that \( \tan(24°) = 1/\tan(66°) \), so the blank should be filled with \( \tan \).

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities, which relate the functions of complementary angles.

Co-function Identity

Co-function identities state that the sine of an angle is equal to the cosine of its complement, and similarly for other functions. For example, sin(θ) = cos(90° - θ) and tan(θ) = cot(90° - θ). This concept is crucial for solving problems that involve angles that add up to 90 degrees, as seen in the given question.

Tangent and Cotangent Functions

The tangent function, tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle, while the cotangent function, cot(θ), is the reciprocal of tangent, or cot(θ) = 1/tan(θ). Understanding the relationship between these functions is vital for filling in the blank in the question, as it requires recognizing that tan(24°) relates to cot(66°) due to their complementary angles.