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Question: How do you solve \(\cos x - 4 = \sin x - 4\) for \(0 \leqslant x \leqslant 2\pi \)?...

How do you solve cos⁡x−4=sin⁡x−4\cos x - 4 = \sin x - 4 for 0⩽x⩽2π0 \leqslant x \leqslant 2\pi ?

Explanation

Solution

First we will evaluate the right-hand of equation and then further the left-hand side of the equation. We will use the relation tan⁡(π4)=1\tan \left( {\dfrac{\pi }{4}} \right) = 1. Then we will try to factorise and simplify the terms so that the left-hand side matches the right-hand side. Then finally evaluate the solution which lies within the given interval.

Complete step by step answer:
We will start solving this question by adding the term 44 to both the sides of the equation.
       cos⁡x−4=sin⁡x−4 cos⁡x−4+4=sin⁡x−4+4  \,\,\,\,\,\,\,\cos x - 4 = \sin x - 4 \\\ \cos x - 4 + 4 = \sin x - 4 + 4 \\\
Now, if we simplify the equation further we will get,
cos⁡x=sin⁡x\cos x = \sin x
Now if we know that it is either cos⁡x=sin⁡x\cos x = \sin x or tan⁡x=1\tan x = 1
We also know that, tan⁡(π4)=1\tan \left( {\dfrac{\pi }{4}} \right) = 1 and tan⁡(π+π4)=1\tan \left( {\pi + \dfrac{\pi }{4}} \right) = 1 or we can say tan⁡(5π4)=1\tan \left( {\dfrac{{5\pi }}{4}} \right) = 1
Hence, the values of xx will be π4,5π4\dfrac{\pi }{4},\dfrac{{5\pi }}{4}.
Now, we know that the given interval is 0⩽x⩽2π0 \leqslant x \leqslant 2\pi for all values of xx.
Since, the terms π4,5π4\dfrac{\pi }{4},\dfrac{{5\pi }}{4} lie within the range [0,2π][0,2\pi ] therefore, both the solutions are feasible.

Hence, the solution of the expression cos⁡x−4=sin⁡x−4\cos x - 4 = \sin x - 4 within the range is π4,5π4\dfrac{\pi }{4},\dfrac{{5\pi }}{4}.

Note: While choosing the side to solve, always choose the side where you can directly apply the trigonometric identities. Also, remember the trigonometric identities sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 and cos⁡2x=2cos⁡2x−1\cos 2x = 2{\cos ^2}x - 1. While applying the double angle identities, first choose the identity according to the terms you have then choose the terms from the expression involving which you are using the double angle identities. While modifying any identity make sure that when you back trace the identity, you get the same original identity. Also, remember that the range of cos⁡\cos function is from −1  - 1\, to +1 + 1 and the range of sin⁡\sin function is also from −1  - 1\, to +1 + 1.