Determine whether each statement is true or false. If false, tell why. Use a calculator for Exercises 39 and 42. sin 42° + sin 42° = sin 84°

Verified step by step guidance

1. Recall the trigonometric identity for the sum of sines: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
2. Apply this identity to the expression \(\sin 42^\circ + \sin 42^\circ\) by setting \(A = 42^\circ\) and \(B = 42^\circ\).
3. Calculate the right side of the identity: \(2 \sin \left( \frac{42^\circ + 42^\circ}{2} \right) \cos \left( \frac{42^\circ - 42^\circ}{2} \right) = 2 \sin 42^\circ \cos 0^\circ\).
4. Since \(\cos 0^\circ = 1\), simplify the expression to \(2 \sin 42^\circ\).
5. Compare \(2 \sin 42^\circ\) with \(\sin 84^\circ\) to determine if the original statement \(\sin 42^\circ + \sin 42^\circ = \sin 84^\circ\) is true or false.

Key Concepts

Sine Addition Formula

The sine addition formula states that sin(A + B) = sin A cos B + cos A sin B. It is used to find the sine of the sum of two angles, which is different from simply adding their sine values. This formula helps verify if sin 42° + sin 42° equals sin 84°.

Properties of Sine Function

The sine function is periodic and nonlinear, meaning sin A + sin B is generally not equal to sin(A + B). Understanding this helps avoid the common mistake of treating sine as a linear operator. Instead, trigonometric identities must be applied for sums.

Use of Calculators for Trigonometric Values

Calculators can compute sine values to verify statements numerically. For example, calculating sin 42°, doubling it, and comparing to sin 84° helps determine the truth of the equation. This practical approach supports theoretical understanding.