In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17

Verified step by step guidance

1. insert step 1> Identify that \( \sin \theta = \frac{15}{17} \) and \( \theta \) is in quadrant II, where sine is positive and cosine is negative.
2. insert step 2> Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).
3. insert step 3> Substitute \( \sin \theta = \frac{15}{17} \) into the identity: \( \left(\frac{15}{17}\right)^2 + \cos^2 \theta = 1 \).
4. insert step 4> Solve for \( \cos^2 \theta \) and then find \( \cos \theta \). Remember that \( \cos \theta \) is negative in quadrant II.
5. insert step 5> Use the double angle formula for cosine: \( \cos 2\theta = 2\cos^2 \theta - 1 \) to find \( \cos 2\theta \).

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 and calculating trigonometric expressions.

Quadrants and Angle Properties

The unit circle is divided into four quadrants, each with distinct properties regarding the signs of sine and cosine. In quadrant II, sine is positive while cosine is negative. This knowledge is essential for determining the values of trigonometric functions based on the angle's location, which directly impacts the calculation of cos(2θ) in this problem.

Finding Exact Values of Trigonometric Functions

To find the exact value of trigonometric functions, one often uses known values or relationships between the functions. Given sin(θ) = 15/17, we can find cos(θ) using the Pythagorean identity sin²(θ) + cos²(θ) = 1. This allows us to compute cos(2θ) accurately by substituting the values derived from sin(θ) and cos(θ) into the double angle formula.