In Exercises 7–14, use the given information to find the exact value of each of the following: b. cos 2θ 24 cos θ = -------- , θ lies in quadrant IV. 25

Verified step by step guidance

1. Step 1: Recall the double angle identity for cosine, which is \( \cos 2\theta = 2\cos^2\theta - 1 \).
2. Step 2: Substitute the given value of \( \cos \theta = \frac{24}{25} \) into the identity.
3. Step 3: Calculate \( \cos^2\theta \) by squaring \( \frac{24}{25} \).
4. Step 4: Substitute \( \cos^2\theta \) into the double angle identity: \( \cos 2\theta = 2\left(\frac{24}{25}\right)^2 - 1 \).
5. Step 5: Simplify the expression to find the exact value of \( \cos 2\theta \).

Key Concepts

Cosine Function

The cosine function, denoted as cos(θ), is a fundamental trigonometric function that relates the angle θ to the ratio of the adjacent side to the hypotenuse in a right triangle. In the context of the unit circle, it represents the x-coordinate of a point on the circle corresponding to the angle θ. Understanding the cosine function is essential for solving problems involving angles and their relationships.

Double Angle Formula

The double angle formula for cosine states that cos(2θ) can be expressed in terms of cos(θ) as cos(2θ) = 2cos²(θ) - 1. This formula allows us to find the cosine of double an angle using the cosine of the original angle, which is particularly useful when the value of cos(θ) is known. Mastery of this formula is crucial for efficiently solving trigonometric problems involving double angles.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to different signs of the trigonometric functions. In quadrant IV, cosine values are positive while sine values are negative. Knowing the quadrant in which an angle lies helps determine the signs of the trigonometric functions, which is vital for accurately calculating values like cos(2θ) when given cos(θ).