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. Recall the reciprocal identity that relates cotangent and tangent: \(\cot \theta = \frac{1}{\tan \theta}\).
2. Substitute the given value of \(\tan \theta = 18\) into the identity: \(\cot \theta = \frac{1}{18}\).
3. Since the denominator is a whole number, check if rationalization is needed. In this case, the denominator is already rational, so no further rationalization is necessary.
4. Express the final answer as a simplified fraction or decimal, depending on the preferred form.
5. Review the result to ensure it aligns with the reciprocal identity and the given value.

Key Concepts

Reciprocal Identities

Reciprocal identities relate pairs of trigonometric functions such as tangent and cotangent, where cot θ is the reciprocal of tan θ. This means cot θ = 1 / tan θ, allowing you to find one function value if the other is known.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers (like square roots) from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression to simplify the expression and present it in a standard form.

Evaluating Trigonometric Functions from Given Values

When given the value of one trigonometric function, you can use identities and algebraic manipulation to find related functions. Here, knowing tan θ allows direct calculation of cot θ using reciprocal identities, ensuring correct substitution and simplification.