Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. cot θ , given that tan θ = 18

Verified step by step guidance

1. Identify the reciprocal identity for cotangent: \( \cot \theta = \frac{1}{\tan \theta} \).
2. Substitute the given value of \( \tan \theta = 18 \) into the reciprocal identity: \( \cot \theta = \frac{1}{18} \).
3. Rationalize the denominator if necessary. In this case, the denominator is already rational, so no further action is needed.
4. Verify that the reciprocal identity is applied correctly by checking that \( \tan \theta \cdot \cot \theta = 1 \).
5. Conclude that the value of \( \cot \theta \) is \( \frac{1}{18} \).

Key Concepts

Reciprocal Identities

Reciprocal identities in trigonometry relate the primary trigonometric functions to their reciprocals. For example, the cotangent function is the reciprocal of the tangent function, expressed as cot(θ) = 1/tan(θ). Understanding these identities is essential for solving problems that require finding one function value based on another.

Tangent Function

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, can also be expressed as tan(θ) = sin(θ)/cos(θ). In this problem, we are given tan(θ) = 18, which allows us to find cot(θ) by applying the reciprocal identity. Recognizing the value of tan(θ) is crucial for determining cot(θ).

Rationalizing Denominators

Rationalizing the denominator is a technique used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will simplify the expression. In trigonometric problems, rationalization may be necessary to present the final answer in a standard form, especially when dealing with reciprocal identities.