Find values of the sine and cosine functions for each angle measure.


θ, given cos 2θ = 3/4 and θ terminates in quadrant III

Verified step by step guidance

1. Step 1: Use the double angle identity for cosine, which is \( \cos 2\theta = 2\cos^2\theta - 1 \).
2. Step 2: Set up the equation \( 2\cos^2\theta - 1 = \frac{3}{4} \) and solve for \( \cos^2\theta \).
3. Step 3: Solve the equation \( 2\cos^2\theta = \frac{7}{4} \) to find \( \cos^2\theta = \frac{7}{8} \).
4. Step 4: Since \( \theta \) is in quadrant III, where cosine is negative, find \( \cos\theta = -\sqrt{\frac{7}{8}} \).
5. Step 5: Use the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) to find \( \sin\theta \), and remember that sine is also negative in quadrant III.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. For any angle θ, the cosine function (cos θ) represents the ratio of the adjacent side to the hypotenuse, while the sine function (sin θ) represents the ratio of the opposite side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and their relationships.

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. These formulas are crucial for finding the sine and cosine values of angles when given information about double angles, as in this problem.

Quadrant Analysis

Quadrant analysis involves understanding the signs of trigonometric functions based on the quadrant in which the angle lies. In quadrant III, both sine and cosine values are negative. This knowledge is vital for determining the correct signs of the sine and cosine values when calculating them from given information, such as cos(2θ) in this case.