Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

Trigonometry

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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2:32 minutes

Problem 69

Textbook Question

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

Step 1: Identify the given function: \( f(\theta) = 2 \cos \theta - \cos 2\theta \).

Step 2: Substitute \( \theta = \frac{\pi}{6} \) into the function: \( f\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{6}\right) - \cos\left(2 \times \frac{\pi}{6}\right) \).

Step 3: Calculate \( \cos\left(\frac{\pi}{6}\right) \). Recall that \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \).

Step 4: Calculate \( \cos\left(2 \times \frac{\pi}{6}\right) \). Simplify \( 2 \times \frac{\pi}{6} = \frac{\pi}{3} \), and recall that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

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.

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Key Concepts

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

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.