Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 7

Chapter 3, Problem 7

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.
$\cos \frac{3 x}{2} \sin \frac{x}{2}$

Verified step by step guidance

Identify the product-to-sum formula that applies to the given expression. The expression is a product of cosine and sine functions with the same angle denominator, so we use the formula: $\cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)]$.

Assign the angles in the expression to variables: let $A = \frac{3x}{2}$ and $B = \frac{x}{2}$.

Substitute $A$ and $B$ into the product-to-sum formula: $\cos \left( \frac{3x}{2} \right) \sin \left( \frac{x}{2} \right) = \frac{1}{2} \left[ \sin \left( \frac{3x}{2} + \frac{x}{2} \right) - \sin \left( \frac{3x}{2} - \frac{x}{2} \right) \right]$.

Simplify the arguments inside the sine functions by combining the fractions: $\frac{3x}{2} + \frac{x}{2} = \frac{4x}{2} = 2x$ and $\frac{3x}{2} - \frac{x}{2} = \frac{2x}{2} = x$.

Rewrite the expression as $\frac{1}{2} [\sin(2x) - \sin(x)]$, which expresses the original product as a difference of sines.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences of trigonometric functions. These identities simplify expressions and solve integrals by rewriting products like cos A sin B as sums, e.g., cos A sin B = 1/2 [sin(A + B) - sin(A - B)].

Trigonometric Function Arguments and Angles

Understanding how to handle angles in trigonometric functions is essential. When applying formulas, correctly identify and manipulate the angles (like x/2) to ensure accurate substitution into identities. This includes recognizing how to add or subtract these angles within the formulas.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves applying identities and algebraic manipulation to rewrite expressions in simpler or more useful forms. This skill is crucial when converting products to sums, as it helps in reducing complex expressions to manageable terms for further analysis or computation.

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Related Practice

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following:a. sin 2θ15sin θ = -------- , θ lies in quadrant II.17

Textbook Question

Use substitution to determine whether the given x-value is a solution of the equation.

$\tan 2 x = - \frac{\sqrt{3}}{3},$

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Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Textbook Question

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

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