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

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Recall the cofunction identity for cotangent: \( \cot(\theta) = \tan\left(\frac{\pi}{2} - \theta\right) \). This means the cotangent of an angle can be expressed as the tangent of its complementary angle.

Identify the given angle: \( \theta = \frac{9\pi}{10} \). We want to express \( \cot\left(\frac{9\pi}{10}\right) \) in terms of a tangent function of a complementary angle.

Calculate the complementary angle to \( \frac{9\pi}{10} \) by subtracting it from \( \frac{\pi}{2} \): \(\n\)\( \frac{\pi}{2} - \frac{9\pi}{10} \).

Simplify the expression for the complementary angle by finding a common denominator and performing the subtraction.

Rewrite \( \cot\left(\frac{9\pi}{10}\right) \) as \( \tan \) of the complementary angle found in the previous step, using the identity from step 1.

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Cofunction Identities

Cofunction identities relate trigonometric functions of complementary angles, where the sum of the angles is π/2 (90°). For example, sine and cosine are cofunctions: sin(θ) = cos(π/2 - θ). These identities help express one trig function in terms of another evaluated at the complementary angle.

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Cofunction Identities

Complementary Angles

Complementary angles are two angles whose measures add up to π/2 radians (90 degrees). Understanding this concept is essential because cofunction identities depend on the relationship between an angle and its complement, allowing transformation of function values accordingly.

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Intro to Complementary & Supplementary Angles

Cotangent Function and Its Cofunction

The cotangent function, cot(θ), is the reciprocal of tangent and is related to the tangent function by cofunction identities. Specifically, cot(θ) = tan(π/2 - θ), meaning cotangent of an angle can be expressed as the tangent of its complementary angle, which is key to rewriting cot(9π/10) in terms of a cofunction.

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Cofunction Identities

Pratica correlata

Domanda del libro di testo

Find the exact value of each expression. See Example 1.

sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .

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Domanda del libro di testo

Find the exact value of each expression.