Write each function value in terms of the cofunction of a complementary angle.
cot (9π/10)

Verified step by step guidance

1. Recognize that cotangent is the cofunction of tangent, and use the identity: \( \cot(\theta) = \tan(\frac{\pi}{2} - \theta) \).
2. Identify the given angle \( \theta = \frac{9\pi}{10} \).
3. Find the complementary angle by calculating \( \frac{\pi}{2} - \frac{9\pi}{10} \).
4. Simplify the expression for the complementary angle.
5. Express \( \cot(\frac{9\pi}{10}) \) as \( \tan(\text{complementary angle}) \).

Key Concepts

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles. For example, the sine of an angle is equal to the cosine of its complement, and vice versa. This means that for any angle θ, sin(θ) = cos(90° - θ) or sin(θ) = cos(π/2 - θ) in radians. Understanding these identities is crucial for rewriting trigonometric functions in terms of their cofunctions.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees (or π/2 radians). In trigonometry, if you have an angle θ, its complement is given by (90° - θ) or (π/2 - θ). Recognizing complementary angles is essential for applying cofunction identities effectively, as it allows for the transformation of trigonometric expressions.

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). It is important to understand how cotangent relates to other trigonometric functions, especially when using cofunction identities. For example, cot(θ) can be expressed in terms of the sine and cosine of complementary angles, which aids in rewriting cot(9π/10) in the context of the problem.