Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

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Recognize that the given expression matches the sine sum identity: \(\sin A \cos B + \cos A \sin B = \sin(A + B)\).

Identify the angles in the expression: \(A = 40^\circ\) and \(B = 20^\circ\).

Rewrite the expression using the identity: \(\sin 40^\circ \cos 20^\circ + \cos 40^\circ \sin 20^\circ = \sin(40^\circ + 20^\circ)\).

Simplify the angle inside the sine function: \(\sin(40^\circ + 20^\circ) = \sin 60^\circ\).

Recall the exact value of \(\sin 60^\circ\) from the unit circle or special triangles, which is \(\frac{\sqrt{3}}{2}\).

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Sum and Difference Identities

Sum and difference identities express the sine, cosine, or tangent of sums or differences of angles in terms of the functions of individual angles. For example, sin(A + B) = sin A cos B + cos A sin B. These identities simplify complex trigonometric expressions and are essential for rewriting expressions like sin 40° cos 20° + cos 40° sin 20°.

Exact Values of Trigonometric Functions

Exact values refer to the precise values of trigonometric functions for special angles, often expressed in terms of square roots and fractions rather than decimals. Knowing exact values for angles like 30°, 45°, and 60° helps in evaluating expressions after applying identities, ensuring answers are accurate and simplified.

Angle Measurement and Conversion

Understanding angle measurement in degrees and radians is crucial for applying trigonometric identities correctly. Recognizing how to manipulate and combine angles, such as adding 40° and 20°, allows for proper use of sum identities and accurate evaluation of the resulting trigonometric function.