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

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Recall the reciprocal identity that relates cotangent and tangent: \(\cot \theta = \frac{1}{\tan \theta}\).

Substitute the given value of \(\tan \theta = 18\) into the identity: \(\cot \theta = \frac{1}{18}\).

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.

Express the final answer as a simplified fraction or decimal, depending on the preferred form.

Review the result to ensure it aligns with the reciprocal identity and the given value.

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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.

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Pythagorean Identities

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.

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Rationalizing Denominators

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.

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