If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

Verified step by step guidance

1. Start with the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\).
2. Substitute \(\sin\theta = \frac{2\sqrt{7}}{7}\) into the identity to get \(\left(\frac{2\sqrt{7}}{7}\right)^2 + \cos^2\theta = 1\).
3. Calculate \(\left(\frac{2\sqrt{7}}{7}\right)^2\) to find \(\sin^2\theta\).
4. Rearrange the equation to solve for \(\cos^2\theta\): \(\cos^2\theta = 1 - \sin^2\theta\).
5. Take the square root of both sides to find \(\cos\theta\), considering that \(\theta\) is an acute angle, so \(\cos\theta\) is positive.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle θ, the relationship sin²θ + cos²θ = 1 holds true. This fundamental identity is derived from the Pythagorean theorem and is essential in trigonometry for relating the sine and cosine of an angle. It allows us to find one trigonometric function if we know the other.

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, we are given sin θ = (2√7) / 7, which is crucial for calculating cos θ using the Pythagorean identity.

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 sine, cosine values for acute angles also range from 0 to 1. By using the Pythagorean identity, we can derive cos θ from the known value of sin θ, allowing us to complete the trigonometric analysis of the angle.