Question

BRAINLIEST IF CORRECT AND EXPLANATION Solve each equation in the interval [0, 2pi)

Answer 1

Answer:  $\frac{\pi}{2}, \ \frac{5\pi}{4}, \ \frac{3\pi}{2}, \ \frac{7\pi}{4}$

We can write the four solutions as:  pi/2, 5pi/4, 3pi/2, 7pi/4

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Work Shown:

$\sqrt{2}\cot(x)\sin(x)+\cot(x)=0\\\\\cot(x)\left(\sqrt{2}\sin(x)+1\right)=0\\\\\cot(x)=0 \ \text{ or } \ \sqrt{2}\sin(x)+1=0\\\\\frac{\cos(x)}{\sin(x)}=0 \ \text{ or } \ \sqrt{2}\sin(x)=-1\\\\\cos(x)=0*\sin(x) \ \text{ or } \ \sin(x)=\frac{-1}{\sqrt{2}}\\\\\cos(x)=0 \ \text{ or } \ \sin(x)=\frac{-\sqrt{2}}{2}$

Use the unit circle or reference sheet to find that $\cos(x)=0$ leads to $x = \frac{\pi}{2} \ \text{ and } \ x = \frac{3\pi}{2}$ when working with the interval [0,2pi)

Also, use of the unit circle would show that $\sin(x)=\frac{-\sqrt{2}}{2}$ leads to $x = \frac{5\pi}{4} \ \text{ and } \ x = \frac{7\pi}{4}$ when working with the interval [0,2pi)

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Overall, the four solutions on the interval [0, 2pi) are

$\frac{\pi}{2}, \ \frac{5\pi}{4}, \ \frac{3\pi}{2}, \ \frac{7\pi}{4}$

To verify these solutions, plug them in one at a time into the original equation. You should end up with 0 = 0 after simplifying. Make sure your calculator is in radian mode.