In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator.If f(θ) = 2 cos θ - cos 2θ, find f(𝜋/6).

Verified step by step guidance

1. Step 1: Identify the given function: \( f(\theta) = 2 \cos \theta - \cos 2\theta \).
2. Step 2: Substitute \( \theta = \frac{\pi}{6} \) into the function: \( f\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{6}\right) - \cos\left(2 \times \frac{\pi}{6}\right) \).
3. Step 3: Calculate \( \cos\left(\frac{\pi}{6}\right) \). Recall that \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \).
4. Step 4: Calculate \( \cos\left(2 \times \frac{\pi}{6}\right) \). Simplify \( 2 \times \frac{\pi}{6} = \frac{\pi}{3} \), and recall that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).
5. Step 5: Substitute the values back into the function: \( f\left(\frac{\pi}{6}\right) = 2 \times \frac{\sqrt{3}}{2} - \frac{1}{2} \). Simplify the expression to express it as a single fraction.

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. It is periodic with a period of 2π and takes values between -1 and 1. Understanding the properties of the cosine function is essential for evaluating expressions involving angles.

Double Angle Formula

The double angle formula for cosine states that cos(2θ) = 2cos²(θ) - 1. This formula allows us to express the cosine of a double angle in terms of the cosine of the original angle, which simplifies calculations. Recognizing and applying this formula is crucial when manipulating trigonometric expressions, especially in problems involving transformations of angles.

Evaluating Trigonometric Functions

Evaluating trigonometric functions at specific angles, such as θ = π/6, involves substituting the angle into the function and using known values of trigonometric functions. For example, cos(π/6) = √3/2. Mastery of these values and the ability to simplify expressions are key skills in solving trigonometric problems accurately.