Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)
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6. Trigonometric Identities and More Equations
Double Angle Identities
Problem 66
Textbook Question
Write each expression as a product of trigonometric functions. See Example 8.
sin 102° - sin 95°
Verified step by step guidance1
Recognize that the expression is a difference of sines: \(\sin 102^\circ - \sin 95^\circ\).
Recall the trigonometric identity for the difference of sines: \(\sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right)\).
Identify \(A = 102^\circ\) and \(B = 95^\circ\) and substitute these values into the identity.
Calculate the average of the angles: \(\frac{102^\circ + 95^\circ}{2}\) and the half difference: \(\frac{102^\circ - 95^\circ}{2}\), but do not simplify the numerical values yet.
Write the expression as a product using the identity: \(2 \cos \left( \frac{102^\circ + 95^\circ}{2} \right) \sin \left( \frac{102^\circ - 95^\circ}{2} \right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum-to-Product Identities
Sum-to-product identities transform sums or differences of sine and cosine functions into products. For example, the difference of sines can be expressed as 2 cos((A+B)/2) sin((A−B)/2). This simplifies expressions and is useful for solving or rewriting trigonometric problems.
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Angle Measurement in Degrees
Trigonometric functions can take angles measured in degrees or radians. Understanding how to work with degrees, including converting or averaging angles, is essential when applying identities like sum-to-product, especially when angles are not standard special angles.
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Basic Sine Function Properties
The sine function is periodic and odd, meaning sin(−θ) = −sin(θ). Recognizing these properties helps in manipulating expressions and applying identities correctly, ensuring accurate transformation of sums or differences of sine terms.
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