Determine whether each statement is true or false. If false, tell why. See Example 4.cos(30° + 60°) = cos 30° + cos 60°

Verified step by step guidance

1. Step 1: Recall the angle addition formula for cosine: \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \).
2. Step 2: Apply the angle addition formula to \( \cos(30^\circ + 60^\circ) \), which becomes \( \cos 30^\circ \cdot \cos 60^\circ - \sin 30^\circ \cdot \sin 60^\circ \).
3. Step 3: Calculate \( \cos 30^\circ \), \( \cos 60^\circ \), \( \sin 30^\circ \), and \( \sin 60^\circ \) using known values: \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), \( \cos 60^\circ = \frac{1}{2} \), \( \sin 30^\circ = \frac{1}{2} \), \( \sin 60^\circ = \frac{\sqrt{3}}{2} \).
4. Step 4: Substitute these values into the formula: \( \cos(30^\circ + 60^\circ) = \frac{\sqrt{3}}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \).
5. Step 5: Compare the result from the angle addition formula with \( \cos 30^\circ + \cos 60^\circ \) to determine if the original statement is true or false.

Key Concepts

Cosine Addition Formula

The cosine addition formula states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B). This formula is essential for calculating the cosine of the sum of two angles, and it shows that the cosine of a sum is not simply the sum of the cosines of the individual angles.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Understanding these identities, such as the sum and difference formulas, is crucial for simplifying expressions and solving trigonometric equations.

Evaluating Trigonometric Functions

Evaluating trigonometric functions at specific angles, such as 30° and 60°, involves knowing their exact values. For example, cos(30°) = √3/2 and cos(60°) = 1/2. This knowledge is necessary to verify the truth of statements involving trigonometric functions.