Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 1.2.69

Chapter 1, Problem 1.2.69

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

Start by substituting \( \theta = \frac{\pi}{6} \) into the function \( f(\theta) = 2 \cos \theta - \cos 2\theta \). This gives \( f\left(\frac{\pi}{6}\right) = 2 \cos \frac{\pi}{6} - \cos \left(2 \times \frac{\pi}{6}\right) \).

Simplify the argument of the second cosine term: \( 2 \times \frac{\pi}{6} = \frac{\pi}{3} \), so the expression becomes \( 2 \cos \frac{\pi}{6} - \cos \frac{\pi}{3} \).

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.

Multiply and combine the terms: \( 2 \times \frac{\sqrt{3}}{2} - \frac{1}{2} \). Simplify the multiplication in the first term.

Express the entire expression as a single fraction by finding a common denominator and combining the terms accordingly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -150°

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In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. csc(7𝜋/6)

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