/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Rewrite each expression as a sim... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 69

Rewrite each expression as a simplified expression containing one term.s $$\cos (\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta$$

Short Answer

Expert verified
The simplified expression is \( \cos\alpha \)

Step by step solution

01

Identify the co-function identities

First of all, recognize the use of sine and cosine identities. The cosine addition formula is \( \cos(A + B) = \cos A \cos B 鈥� \sin A \sin B \), and the sine addition formula is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). The given expression can be seen as a combination of these two formulas.
02

Rearrange and substitute

Reverse the given expression to align with the formula. It then becomes \( \cos\beta \cos(\alpha + \beta) + \sin\beta \sin(\alpha + \beta) \). Now, substitute \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \) with the co-function identities. This leads to \[ \cos\beta [\cos\alpha \cos\beta - \sin\alpha \sin\beta] + \sin\beta [\sin\alpha \cos\beta + \cos\alpha \sin\beta] \]
03

Simplify the expression

Simplify this expression. After the multiplication, the result is \( \cos^{2}\beta \cos\alpha - \cos\beta \sin\alpha \sin\beta + \cos\beta \sin\alpha \sin\beta + \cos\alpha \sin^{2}\beta \). Observe that \( -\cos\beta \sin\alpha \sin\beta \) and \( \cos\beta \sin\alpha \sin\beta \) cancel each other out because they are same in magnitude but opposite in sign. As a result, the expression simplifies to \( \cos^{2}\beta \cos\alpha + \cos\alpha \sin^{2}\beta \).
04

Factor out common terms

Factor out the common term \( \cos\alpha \) from the simplified equation to get \[ \cos\alpha (\cos^{2}\beta + \sin^{2}\beta) \]. This can be further simplified because \( \cos^{2}\beta + \sin^{2}\beta = 1 \) (known as the Pythagorean trigonometric identity). The final simplified expression is \( \cos\alpha \).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigonometric expression.

Verify the identity: $$\frac{\sin (x-y)}{\cos x \cos y}+\frac{\sin (y-z)}{\cos y \cos z}+\frac{\sin (z-x)}{\cos z \cos x}=0$$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos x \csc x=2 \cos x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A trigonometric equation with an infinite number of solutions is an identity.
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