In Exercises 7–14, use the given information to find the exact value of each of the following: b. cos 2θ 2 sin θ = ﹣ -------- , θ lies in quadrant III. 3

Key Concepts

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For cosine, the formula is cos(2θ) = cos²(θ) - sin²(θ) or alternatively, cos(2θ) = 2cos²(θ) - 1. Understanding these formulas is essential for calculating values like cos(2θ) when given sin(θ) or cos(θ).

Trigonometric Values in Quadrants

The unit circle defines the signs of trigonometric functions in different quadrants. In quadrant III, both sine and cosine values are negative. This is crucial for determining the correct signs of sin(θ) and cos(θ) when calculating cos(2θ) based on the given information about θ's location.

Pythagorean Identity

The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This identity allows us to find the cosine value when we know the sine value, especially useful in this problem where sin(θ) is provided. By rearranging the identity, we can derive cos(θ) and subsequently use it to find cos(2θ).