Find the exact value of each expression. See Example 1.
sin 40° cos 50° + cos 40° sin 50°

Verified step by step guidance

1. Recognize that the expression \( \sin 40^\circ \cos 50^\circ + \cos 40^\circ \sin 50^\circ \) matches the sine addition formula, which states: \( \sin A \cos B + \cos A \sin B = \sin(A + B) \).
2. Identify the angles \( A = 40^\circ \) and \( B = 50^\circ \) from the given expression.
3. Apply the sine addition formula by substituting the values of \( A \) and \( B \): \( \sin(40^\circ + 50^\circ) \).
4. Simplify the angle inside the sine function: \( 40^\circ + 50^\circ = 90^\circ \).
5. Recall the exact value of \( \sin 90^\circ \), which is a well-known trigonometric value.

Key Concepts

Sum of Angles Identity for Sine

The sum of angles identity states that sin(A + B) = sin A cos B + cos A sin B. This formula allows the expression sin 40° cos 50° + cos 40° sin 50° to be rewritten as sin(40° + 50°), simplifying the calculation.

Exact Values of Trigonometric Functions

Exact values refer to the precise trigonometric values for special angles, often expressed in terms of square roots or fractions. While 40° and 50° are not standard special angles, using identities can reduce expressions to angles whose sine or cosine values are easier to interpret or approximate.

Angle Measurement in Degrees

Trigonometric functions can be evaluated in degrees or radians. Here, the angles are given in degrees, so it is important to ensure that calculations and identities are applied consistently using degree measure to avoid errors.