/*! 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 12 Verify each identity. $$\cos \... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 12

Verify each identity. $$\cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

Short Answer

Expert verified
The given expression \(\cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)\) is indeed a valid trigonometric identity as per the verification process involving usage of the formula for cosine difference and simplification.

Step by step solution

01

Recall the formula for cosine of difference of two angles

The formula for cosine of difference of two angles (cos (A-B)) is given by:\[cos(A-B) = cosA cosB + sinA sinB\]
02

Apply the potential trigonometric identity

Start with the left side of the given potential trigonometric identity, \[\cos \left(x-\frac{5 \pi}{4}\right)\] and apply the formula from step 1.\[\cos \left(x-\frac{5 \pi}{4}\right) = \cos x \cos \left(\frac{5 \pi}{4}\right) + \sin x \sin \left(\frac{5 \pi}{4}\right)\]
03

Evaluate the trigonometric functions

The cos(5蟺/4) and sin(5蟺/4) are each equal to -鈭�2/2. Substitute these values in the equation obtained in step 2.\[\cos \left(x-\frac{5 \pi}{4}\right) = \cos x \left(-\frac{\sqrt{2}}{2}\right) + \sin x \left(-\frac{\sqrt{2}}{2}\right)\]
04

Simplify the obtained expression

Now simplify the expression obtained in step 3 by factoring out -鈭�2/2.\[\cos (x - \frac{5 \pi}{4}) = -\frac{\sqrt{2}}{2} (\cos x + \sin x)\] which is the right side of the identity. This shows that the given formula is an identity, thus completing the verification process.

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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. $$\frac{\sin x}{1-\cos ^{2} x}=\csc x$$

Describe a natural periodic phenomenon. Give an example of a question that can be answered by a trigonometric equation in the study of this phenomenon.

Use this information to solve. Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, \(y,\) measured in liters per second, after \(x\) seconds is modeled by $$y=0.6 \sin \frac{2 \pi}{5} x$$ Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we exhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.

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

Determine the amplitude and period of \(y=3 \cos 2 \pi x\) Then graph the function for \(-4 \leq x \leq 4\) (Section 4.5, Example 5)
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