In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ12sin θ = -------- , θ lies in quadrant II.13

Verified step by step guidance

1. Step 1: Recall the double angle identity for cosine. The formula is \( \cos 2\theta = 1 - 2\sin^2 \theta \) or \( \cos 2\theta = 2\cos^2 \theta - 1 \).
2. Step 2: Since \( \sin \theta = \frac{12}{13} \), use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).
3. Step 3: Substitute \( \sin \theta = \frac{12}{13} \) into the Pythagorean identity to find \( \cos^2 \theta \). Calculate \( \cos^2 \theta = 1 - \left(\frac{12}{13}\right)^2 \).
4. Step 4: Solve for \( \cos \theta \). Since \( \theta \) is in quadrant II, \( \cos \theta \) is negative.
5. Step 5: Use the value of \( \cos \theta \) found in Step 4 to calculate \( \cos 2\theta \) using the double angle identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One key identity is the double angle formula for cosine, which states that cos(2θ) can be expressed as cos²(θ) - sin²(θ) or 2cos²(θ) - 1 or 1 - 2sin²(θ). Understanding these identities is crucial for simplifying expressions and solving trigonometric equations.

Quadrants and Angle Properties

The unit circle is divided into four quadrants, each with specific signs for sine and cosine values. In quadrant II, sine is positive while cosine is negative. Knowing the quadrant in which an angle lies helps determine the signs of the trigonometric functions, which is essential for accurately calculating values like cos(2θ) when given sin(θ).

Finding Exact Values

Finding exact values of trigonometric functions often involves using known values from special angles (like 30°, 45°, and 60°) or applying identities. In this case, with sin(θ) given, one can find cos(θ) using the Pythagorean identity sin²(θ) + cos²(θ) = 1. This allows for the calculation of cos(2θ) using the derived value of cos(θ) along with the double angle formula.