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Chapter 5: Problem 81

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x+\frac{\pi}{4}\right)-\cos \left(x-\frac{\pi}{4}\right)=1$$

Short Answer

Expert verified
The solutions of the given equation in the interval \([0,2 \pi)\) are \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\).

Step by step solution

01

Simplify equation using trigonometric identity

Apply the 'cos difference identity' which is \( cos(a) - cos(b) = -2sin(\frac{a+b}{2})sin(\frac{a-b}{2}) \). The equation thus becomes: \(-2sin(\frac{2x}{2})sin(-\frac{\pi}{2}) = 1\).
02

Simplify Further

The equation further simplifies to \(2sin(x) = 1\), as sine of \(-\frac{\pi}{2}\) equals to -1.
03

Solve for x

Solve for x by taking the inverse sine on both sides gives, \(x = sin^{-1}(0.5)\). As we are looking for all solutions in the given interval \([0,2 \pi)\) which means \([0, 360^{\circ}]\) in degree, we have to consider that sine function is positive in the 1st and 2nd quadrant. Hence, we get \(x_1 = sin^{-1}(0.5) = 30^{\circ}\) and \(x_2 = 180^{\circ} - x_1 = 180^{\circ} - 30^{\circ} = 150^{\circ}\).
04

Convert back to radians

To present the final answer in radian (since the solutions should be given in the interval \([0,2\pi)\), which is in radian), convert the degrees into radians. This gives \(x_1 = \frac{\pi}{6}\) radian and \(x_2 = \frac{5\pi}{6}\) radian.

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Most popular questions from this chapter

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\csc ^{2} x+3 \csc x-4=0$$

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Fill in the blank. \(\sin (u+v)=\)_____
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