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. Recall the cosine addition formula: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
2. Apply the formula to \(\cos(30^\circ + 60^\circ)\), which becomes \(\cos 30^\circ \cos 60^\circ - \sin 30^\circ \sin 60^\circ\).
3. Compare this expression to the right side of the given statement, which is \(\cos 30^\circ + \cos 60^\circ\).
4. Notice that the given statement adds the cosines directly, but the correct formula involves products of cosines and sines with a subtraction.
5. Conclude that the statement is false because it does not follow the cosine addition identity; the correct expression includes both cosine and sine terms multiplied and subtracted.

Key Concepts

Cosine Addition Formula

The cosine addition formula states that cos(A + B) = cos A cos B - sin A sin B. It is used to find the cosine of the sum of two angles, which is not simply the sum of their cosines. This formula is essential to verify or refute the given statement.

Properties of Trigonometric Functions

Trigonometric functions like cosine and sine have specific properties and identities that govern their behavior. Understanding that cosine is not a linear function and that cos(A + B) ≠ cos A + cos B helps in evaluating the truth of the statement.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting known angle values and simplifying using identities. For example, calculating cos(30° + 60°) and comparing it to cos 30° + cos 60° helps determine if the statement is true or false.