In Exercises 31–38, find a cofunction with the same value as the given expression.sin 7°

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1. Understand that cofunctions are pairs of trigonometric functions that are equal when their angles are complementary. The cofunction identity for sine is: \( \sin(\theta) = \cos(90^\circ - \theta) \).
2. Identify the given angle \( \theta = 7^\circ \).
3. Use the cofunction identity for sine: \( \sin(7^\circ) = \cos(90^\circ - 7^\circ) \).
4. Calculate the complementary angle: \( 90^\circ - 7^\circ = 83^\circ \).
5. Conclude that the cofunction with the same value as \( \sin(7^\circ) \) is \( \cos(83^\circ) \).

Key Concepts

Cofunction Identities

Cofunction identities in trigonometry relate the values of trigonometric functions of complementary angles. For example, the sine of an angle is equal to the cosine of its complement: sin(θ) = cos(90° - θ). This relationship is crucial for finding cofunctions that yield the same value for given angles.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In the context of trigonometric functions, knowing the complementary angle allows us to apply cofunction identities effectively. For instance, if we have sin(7°), its complementary angle is 90° - 7° = 83°.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in relating angles to side lengths in right triangles. Understanding these functions and their properties is essential for solving problems involving angles and their relationships, including finding cofunctions.