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 = 6/7

Verified step by step guidance

1. Start with the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
2. Substitute the given value of \( \sin \theta = \frac{6}{7} \) into the identity: \( \left( \frac{6}{7} \right)^2 + \cos^2 \theta = 1 \).
3. Calculate \( \sin^2 \theta \) by squaring \( \frac{6}{7} \): \( \sin^2 \theta = \frac{36}{49} \).
4. Rearrange the equation to solve for \( \cos^2 \theta \): \( \cos^2 \theta = 1 - \frac{36}{49} \).
5. Simplify the expression to find \( \cos^2 \theta \), then take the square root to find \( \cos \theta \). Remember, since \( \theta \) is an acute angle, \( \cos \theta \) will be positive.

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 derive one from 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. In this context, sin θ = 6/7 indicates that for an acute angle θ, the opposite side is 6 units long while the hypotenuse is 7 units long. Understanding this ratio is crucial for applying the Pythagorean identity effectively.

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. By using the Pythagorean identity, we can find cos θ when sin θ is known. In this case, once we calculate sin²θ, we can rearrange the identity to solve for cos²θ, leading to the determination of cos θ.