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°

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Identify the trigonometric identity or property that might apply to the given equation.

Recall that \( \sin(A + B) \neq \sin A + \sin B \) in general, which means the sum of sines is not equal to the sine of the sum.

Calculate \( \sin 42^\circ + \sin 42^\circ \) using the fact that it simplifies to \( 2 \sin 42^\circ \).

Calculate \( \sin 84^\circ \) separately using a calculator.

Compare the values of \( 2 \sin 42^\circ \) and \( \sin 84^\circ \) to determine if the original statement is true or false.

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Key Concepts

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

Sine Function

The sine function is a fundamental trigonometric function defined for an angle in a right triangle as the ratio of the length of the opposite side to the hypotenuse. It is also defined on the unit circle, where the sine of an angle corresponds to the y-coordinate of the point on the circle. Understanding the sine function is crucial for evaluating expressions involving angles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the sine addition formula, which states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Recognizing and applying these identities is essential for simplifying trigonometric expressions and solving equations.

Angle Addition in Sine

The angle addition property for sine states that sin(a + b) is not equal to sin(a) + sin(b). Instead, it requires the use of the sine addition formula. This concept is critical for understanding why the statement sin 42° + sin 42° does not equal sin 84°, as it illustrates the need to apply the correct trigonometric identities when combining angles.

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