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Chapter 5: Problem 34

Verify each identity. $$\sin \left(x+\frac{3 \pi}{2}\right)=-\cos x$$

Short Answer

Expert verified
The identity is correct.

Step by step solution

01

Recognize the Sum Identity for Sine

The Sum Identity for sine states that for any two angles \(A\) and \(B\), \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). We can apply this to our equation by letting \(A = x\) and \(B = \frac{3\pi}{2}\) . Thus, \(\sin \left(x + \frac{3 \pi}{2}\right) = \sin x \cos \frac{3 \pi}{2} + \cos x \sin \frac{3 \pi}{2}\).
02

Evaluate the Trigonometric Functions

We know that \(\cos \frac{3\pi}{2} = 0\) and \(\sin \frac{3\pi}{2} = -1\). Substituting these into the equation gives us the result \(= \sin x * 0 + \cos x * (-1)\). This simplifies to \(-\cos x\).
03

Compare both Sides

We can now clearly see that both sides of our original equation are equal: \(\sin \left(x + \frac{3 \pi}{2}\right) = -\cos x\). Therefore, we have verified the given identity.

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Most popular questions from this chapter

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin \left(x+\frac{\pi}{4}\right)=\sin x+\sin \frac{\pi}{4}$$

Find the exact value of each expression. Do not use a calculator. $$\sin \left(\cos ^{-1} \frac{1}{2}+\sin ^{-1} \frac{3}{5}\right)$$

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. $$\cos x \csc x=2 \cos x$$

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. $$5 \sin x=2 \cos ^{2} x-4$$

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. $$\tan x=-4.7143$$
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