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. Start by substituting \( \theta = \frac{\pi}{6} \) into the function \( f(\theta) = 2 \cos \theta - \cos 2\theta \). This gives \( f\left(\frac{\pi}{6}\right) = 2 \cos \frac{\pi}{6} - \cos \left(2 \times \frac{\pi}{6}\right) \).
2. Simplify the argument of the second cosine term: \( 2 \times \frac{\pi}{6} = \frac{\pi}{3} \), so the expression becomes \( 2 \cos \frac{\pi}{6} - \cos \frac{\pi}{3} \).
3. Recall the exact values of cosine for these special angles: \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \). Substitute these values into the expression.
4. Multiply and combine the terms: \( 2 \times \frac{\sqrt{3}}{2} - \frac{1}{2} \). Simplify the multiplication in the first term.
5. Express the entire expression as a single fraction by finding a common denominator and combining the terms accordingly.

Key Concepts

Evaluating Trigonometric Functions at Specific Angles

To find the value of a trigonometric function at a given angle, substitute the angle into the function and use known exact values of sine and cosine for common angles like π/6. This avoids approximation and ensures exact results.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of 2θ in terms of θ, such as cos 2θ = 2 cos² θ - 1. These formulas simplify expressions and are essential for rewriting functions like f(θ) = 2 cos θ - cos 2θ in terms of a single angle.

Expressing Results as a Single Fraction

After evaluating and simplifying trigonometric expressions, combine terms into a single fraction to present the exact value neatly. This involves finding a common denominator and simplifying numerator and denominator without decimal approximations.