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Chapter 5: Problem 36

Verify each identity. $$\cos (\pi-x)=-\cos x$$

Short Answer

Expert verified
After evaluating the identity using the cosine of difference formula, we find that indeed \(\cos (\pi-x)=-\cos x\).

Step by step solution

01

Setup

We are asked to prove, for any angle \(x\), that \(\cos (\pi-x)=-\cos x\).
02

Using the cosine of difference identity

We know that the cosine of difference of two angles is given by \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). So we can rewrite our expression as \(cos(\pi - x) = \cos\pi \cos x + \sin\pi \sin x\).
03

Evaluate

We know that \(\cos\pi = -1\) and \(\sin\pi = 0\). So our expression becomes \(-1 \cdot \cos x + 0 = -\cos x\).

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Most popular questions from this chapter

Exercises \(116-118\) will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$\sin 60^{\circ} \sin 30^{\circ}=\frac{1}{2}\left[\cos \left(60^{\circ}-30^{\circ}\right)-\cos \left(60^{\circ}+30^{\circ}\right)\right]$$

Exercises \(116-118\) will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$\sin \pi \cos \frac{\pi}{2}=\frac{1}{2}\left[\sin \left(\pi+\frac{\pi}{2}\right)+\sin \left(\pi-\frac{\pi}{2}\right)\right]$$

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. $$5 \sec ^{2} x-10=0$$

Solve: \(\log x+\log (x+1)=\log 12\) (Section 3.4, Example 8)

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.
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