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 5.34

Textbook Question

Use identities to fill in each blank with the appropriate trigonometric function name.
____ 72° = cot 18°

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: \( \sin(90° - \theta) = \cos(\theta) \), \( \cos(90° - \theta) = \sin(\theta) \), \( \tan(90° - \theta) = \cot(\theta) \), and \( \cot(90° - \theta) = \tan(\theta) \).

Identify that 72° and 18° are complementary angles because 72° + 18° = 90°.

Apply the identity for cotangent: \( \cot(90° - \theta) = \tan(\theta) \).

Conclude that \( \tan(72°) = \cot(18°) \).

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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 different trigonometric functions to one another.

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 pairs of trigonometric functions. For example, sin(θ) = cos(90° - θ) and cot(θ) = tan(90° - θ). This concept is crucial for solving problems that involve angles that add up to 90 degrees, such as the relationship between cotangent and tangent.

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = 1/tan(θ) or cot(θ) = cos(θ)/sin(θ). It is important in trigonometry for solving right triangles and in various applications, including calculus and physics. Understanding cotangent helps in recognizing relationships between angles and their corresponding trigonometric values.