In Exercises 31–38, find a cofunction with the same value as the given expression.cos 2𝜋5

Verified step by step guidance

1. Identify the given trigonometric function: \( \cos \left( \frac{2\pi}{5} \right) \).
2. Recall the cofunction identity: \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \).
3. Substitute \( \theta = \frac{2\pi}{5} \) into the cofunction identity: \( \cos\left(\frac{2\pi}{5}\right) = \sin\left(\frac{\pi}{2} - \frac{2\pi}{5}\right) \).
4. Simplify the expression inside the sine function: \( \frac{\pi}{2} - \frac{2\pi}{5} \).
5. The cofunction with the same value is \( \sin\left(\frac{\pi}{2} - \frac{2\pi}{5}\right) \).

Key Concepts

Cofunction Identities

Cofunction identities in trigonometry relate the values of trigonometric functions of complementary angles. For example, the sine of an angle is equal to the cosine of its complement, and vice versa. This means that sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). Understanding these identities is crucial for finding cofunctions that yield the same value.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and the coordinates of points on a circle with a radius of one. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine and the y-coordinate represents the sine of that angle. Familiarity with the unit circle helps in visualizing and calculating trigonometric values, including cofunctions.

Periodic Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For cosine, the period is 2π, which indicates that cos(θ) = cos(θ + 2πn) for any integer n. This property is essential when evaluating expressions like cos(2π/5), as it allows for the identification of equivalent angles and their corresponding cofunction values.