Textbook Question
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°
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6. Trigonometric Identities and More Equations
Solving Trigonometric Equations Using Identities
3:09 minutesProblem 3.RE.45
Textbook Question
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
Verified step by step guidance1
Recall the sine difference identity for expressing the difference of sines as a product: \(\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)\).
Identify \(A\) and \(B\) in the expression \(\sin 2x - \sin 4x\) as \(A = 2x\) and \(B = 4x\).
Apply the formula by substituting \(A\) and \(B\): \(\sin 2x - \sin 4x = 2 \cos \left( \frac{2x + 4x}{2} \right) \sin \left( \frac{2x - 4x}{2} \right)\).
Simplify the arguments inside the cosine and sine functions: \(\cos \left( 3x \right)\) and \(\sin \left( -x \right)\).
Use the odd property of sine, \(\sin(-x) = -\sin x\), to rewrite the expression as a product involving \(\cos 3x\) and \(\sin x\).
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Video duration:
3mKey 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 sine differences, the identity is sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2). This simplifies expressions and helps in solving or evaluating trigonometric problems.
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Angle Substitution and Simplification
After applying identities, substituting the given angles correctly is crucial. Simplifying expressions like (2x + 4x)/2 and (2x - 4x)/2 ensures accurate transformation. This step is essential for reducing the expression to a product form.
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Exact Values of Trigonometric Functions
Finding the exact value of a trigonometric expression involves knowing standard angle values and their sine or cosine results. For example, angles like 0°, 30°, 45°, 60°, and 90° have well-known exact sine and cosine values, which help evaluate the product precisely.
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