Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(θ/2)
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
2:07 minutesProblem 13
Textbook Question
Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 4x + cos 2x
Verified step by step guidance1
Recognize that the expression is a sum of two cosine functions: \(\cos 4x + \cos 2x\).
Recall the sum-to-product identity for cosine sums: \(\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
Identify \(A = 4x\) and \(B = 2x\), then substitute these into the formula: \(\cos 4x + \cos 2x = 2 \cos \left( \frac{4x + 2x}{2} \right) \cos \left( \frac{4x - 2x}{2} \right)\).
Simplify the arguments inside the cosine functions: \(\frac{4x + 2x}{2} = 3x\) and \(\frac{4x - 2x}{2} = x\).
Write the expression as a product: \(2 \cos 3x \cos x\). If needed, you can then evaluate this product for specific values of \(x\) to find the exact value.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum-to-Product Formulas
Sum-to-product formulas transform sums or differences of trigonometric functions into products, simplifying expressions and solving equations. For example, the sum of cosines can be expressed as a product: cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). This is essential for rewriting cos 4x + cos 2x as a product.
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Trigonometric Identities
Trigonometric identities are equations involving trig functions that hold true for all values in their domains. They allow manipulation and simplification of expressions. Knowing identities like angle addition, subtraction, and double-angle formulas helps in recognizing patterns and applying sum-to-product transformations effectively.
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Exact Values of Trigonometric Functions
Exact values refer to precise trigonometric values for special angles (e.g., 0°, 30°, 45°, 60°, 90°) often expressed in radicals or fractions. After expressing sums as products, evaluating the product's exact value requires familiarity with these standard angles and their trig values to provide precise answers without decimal approximations.
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