Find values of the sine and cosine functions for each angle measure.
2θ, given cos θ = -12/13 and sin θ > 0

Verified step by step guidance

1. Recognize that you are given \( \cos \theta = -\frac{12}{13} \) and \( \sin \theta > 0 \). This indicates that \( \theta \) is in the second quadrant, where cosine is negative and sine is positive.
2. Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). Substitute \( \cos \theta = -\frac{12}{13} \) into the identity: \( \sin^2 \theta + \left(-\frac{12}{13}\right)^2 = 1 \).
3. Solve for \( \sin \theta \) by calculating \( \sin^2 \theta = 1 - \left(\frac{144}{169}\right) \) and then take the positive square root since \( \sin \theta > 0 \).
4. To find \( \sin 2\theta \) and \( \cos 2\theta \), use the double angle formulas: \( \sin 2\theta = 2 \sin \theta \cos \theta \) and \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
5. Substitute the values of \( \sin \theta \) and \( \cos \theta \) into the double angle formulas to find \( \sin 2\theta \) and \( \cos 2\theta \).

Key Concepts

Trigonometric Functions

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

Double Angle Formulas

Double angle formulas are used to express trigonometric functions of double angles in terms of single angles. For sine, the formula is sin(2θ) = 2sin(θ)cos(θ), and for cosine, it is cos(2θ) = cos²(θ) - sin²(θ). These formulas are crucial for finding the sine and cosine of angles that are multiples of a given angle.

Quadrants and Sign of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of the sine and cosine functions. In this case, cos θ is negative and sin θ is positive, indicating that θ is in the second quadrant. Understanding the signs of these functions in different quadrants is vital for accurately determining their values for specific angles.