Textbook Question
In Exercises 57–64, find the exact value of the following under the given conditions:
a. cos (α + β)
tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
2:17 minutesProblem 1
Textbook Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.sin 6x sin 2x
Verified step by step guidance1
Identify the trigonometric identity that can be used to express the product \( \sin A \sin B \) as a sum or difference. The relevant identity is: \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \).
Substitute \( A = 6x \) and \( B = 2x \) into the identity: \( \sin 6x \sin 2x = \frac{1}{2} [\cos(6x - 2x) - \cos(6x + 2x)] \).
Simplify the expressions inside the cosine functions: \( \cos(6x - 2x) = \cos(4x) \) and \( \cos(6x + 2x) = \cos(8x) \).
Substitute the simplified expressions back into the equation: \( \sin 6x \sin 2x = \frac{1}{2} [\cos(4x) - \cos(8x)] \).
The expression \( \sin 6x \sin 2x \) is now written as a sum or difference of cosines: \( \frac{1}{2} [\cos(4x) - \cos(8x)] \).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product-to-Sum Formulas
Product-to-sum formulas are trigonometric identities that allow the transformation of products of sine and cosine functions into sums or differences. For example, the formula for sin(A)sin(B) is given by sin(A)sin(B) = 1/2 [cos(A-B) - cos(A+B)]. These formulas simplify calculations and are essential for solving problems involving products of trigonometric functions.
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Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved. They include fundamental identities such as the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for manipulating and simplifying trigonometric expressions, which is often necessary in solving problems in trigonometry.
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Angle Addition and Subtraction
Angle addition and subtraction formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For instance, sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B). These formulas are vital for breaking down complex trigonometric expressions into simpler components, facilitating easier calculations and problem-solving.
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