Determine whether each statement is true or false. If false, tell why. See Example 4.cos 60° = 2 cos² 30° - 1

Verified step by step guidance

1. Step 1: Recall the value of \( \cos 60^\circ \). It is known that \( \cos 60^\circ = \frac{1}{2} \).
2. Step 2: Use the double angle identity for cosine, which states \( \cos 2\theta = 2\cos^2 \theta - 1 \).
3. Step 3: Substitute \( \theta = 30^\circ \) into the double angle identity: \( \cos 60^\circ = 2\cos^2 30^\circ - 1 \).
4. Step 4: Recall the value of \( \cos 30^\circ \). It is known that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \).
5. Step 5: Substitute \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) into the equation from Step 3 and simplify to verify if both sides are equal.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all angles and is periodic, with a range of values between -1 and 1. Understanding the cosine function is essential for evaluating expressions involving angles, such as cos 60°.

Double Angle Formula

The double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For cosine, the formula is cos(2θ) = 2cos²(θ) - 1. This identity is crucial for simplifying expressions and solving equations involving trigonometric functions, particularly when verifying statements like the one in the question.

Evaluating Trigonometric Values

Evaluating trigonometric values involves calculating the sine, cosine, or tangent of specific angles, often using known values from the unit circle or trigonometric tables. For example, cos 60° equals 0.5, and cos 30° equals √3/2. This skill is necessary to determine the truth of statements involving trigonometric identities and to provide accurate justifications for any false claims.