In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ.__sin θ = √398

Verified step by step guidance

1. Recognize that you are given \( \sin \theta = \frac{\sqrt{39}}{8} \) and need to find \( \cos \theta \).
2. Use the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
3. Substitute \( \sin \theta = \frac{\sqrt{39}}{8} \) into the identity: \( \left(\frac{\sqrt{39}}{8}\right)^2 + \cos^2 \theta = 1 \).
4. Calculate \( \left(\frac{\sqrt{39}}{8}\right)^2 \) to find \( \sin^2 \theta \).
5. Solve for \( \cos^2 \theta \) by subtracting \( \sin^2 \theta \) from 1, then take the square root to find \( \cos \theta \).

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin²θ + cos²θ = 1 holds true. This fundamental identity connects the sine and cosine functions, allowing us to find one if we know the other. It is particularly useful in trigonometry for solving problems involving right triangles and circular functions.

Sine Function

The sine function, denoted as sin θ, represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. For acute angles, the sine value is always positive and ranges from 0 to 1. In this problem, sin θ is given as √39/8, which indicates the relationship between the angle and the sides of the triangle.

Cosine Function

The cosine function, denoted as cos θ, represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Like the sine function, the cosine of an acute angle is also positive and ranges from 0 to 1. By using the Pythagorean identity, we can calculate cos θ when sin θ is known, providing a complete understanding of the angle's trigonometric properties.