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Chapter 5: Problem 32

Factor the trigonometric expression. There is more than one correct form of each answer. $$\sin ^{2} x+3 \cos x+3$$

Short Answer

Expert verified
The factorized expression is \( -(\sin x - \sqrt{1 - \cos^2 x} -2)(\sin x - \sqrt{1 - \cos^2 x} - 2) \)

Step by step solution

01

Recognition of trigonometric identities

The problem presents a quadratic-like expression with \( \sin ^{2} x \), \( \cos x \), and a constant term. Note that \( \sin ^{2} x \) can be replaced with \( 1 - \cos ^{2} x \) using the Pythagorean identity in trigonometry, so our equation becomes \( 1 - \cos ^{2} x + 3 \cos x + 3 \).
02

Rearrange into standard quadratic form

Rearrange the equation from step 1 to mirror a quadratic in the form of \( ax^{2} + bx + c \). This gives us: \( - \cos ^{2} x + 3 \cos x + 4 \).
03

Factor the quadratic

Factor the quadratic: \( -(\cos x -2)(\cos x - 2) \)
04

Finalize the factorization

Re-substitute \(\cos x\) to give the final answer. This leaves us with \( -(\sin x - \sqrt{1 - \cos^2 x} -2)(\sin x - \sqrt{1 - \cos^2 x} - 2) \)

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Most popular questions from this chapter

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