In Exercises 31–38, find a cofunction with the same value as the given expression. tan 𝜋 9

Verified step by step guidance

1. Recall the cofunction identity for tangent: \(\tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right)\), where cotangent is the cofunction of tangent.
2. Identify the given angle \(\theta = \frac{\pi}{9}\) in the expression \(\tan\left(\frac{\pi}{9}\right)\).
3. Apply the cofunction identity by substituting \(\theta\) into the formula: \(\tan\left(\frac{\pi}{9}\right) = \cot\left(\frac{\pi}{2} - \frac{\pi}{9}\right)\).
4. Simplify the angle inside the cotangent: \(\frac{\pi}{2} - \frac{\pi}{9} = \frac{9\pi}{18} - \frac{2\pi}{18} = \frac{7\pi}{18}\).
5. Write the final cofunction expression: \(\tan\left(\frac{\pi}{9}\right) = \cot\left(\frac{7\pi}{18}\right)\).

Key Concepts

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, such as sin(θ) = cos(90° - θ) or tan(θ) = cot(90° - θ). These identities help find equivalent expressions by using the complementary angle concept, where the sum of angles is 90° (or π/2 radians).

Radian Measure and Conversion

Angles can be measured in degrees or radians; understanding radian measure is essential for working with trigonometric functions. π radians equal 180°, so converting between radians and degrees helps interpret and apply cofunction identities correctly.

Tangent and Cotangent Functions

Tangent (tan) and cotangent (cot) are reciprocal trigonometric functions, where cot(θ) = 1/tan(θ). Recognizing their relationship is key to finding cofunctions with the same value, especially when using complementary angle identities involving tan and cot.