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

How do you solve sin⁡2x−sin⁡x=0{\sin ^2}x - \sin x = 0 for 0⩽x⩽2π0 \leqslant x \leqslant 2\pi ?

Explanation

Solution

In this question, we want to find the trigonometry angle value of the given equation between the intervals 0 to 2π2\pi . Apply the formula sin⁡2x=2sin⁡xcos⁡x{\sin ^2}x = 2\sin x\cos x
Use the factorization method to solve the equation. At the end, we will find the value of the function. Based on those values, we will be able to find the value of the angle.

Complete step-by-step answer:
In this question, given that
⇒sin⁡2x−sin⁡x=0\Rightarrow {\sin ^2}x - \sin x = 0
As we already know that,
sin⁡2x=2sin⁡xcos⁡x{\sin ^2}x = 2\sin x\cos x
Let us substitute the values in the given equation.
⇒2sin⁡xcos⁡x+sin⁡x=0\Rightarrow 2\sin x\cos x + \sin x = 0
Let us take out the common factor.
⇒sin⁡x(2cos⁡x+1)=0\Rightarrow \sin x\left( {2\cos x + 1} \right) = 0
Now, equate both the factors to zero to obtain the solution.
For the first factor:
⇒sin⁡x=0\Rightarrow \sin x = 0
And for the second factor:
⇒2cos⁡x+1=0\Rightarrow 2\cos x + 1 = 0
Let us do subtraction by -1 on both sides.
⇒2cos⁡x+1−1=0−1\Rightarrow 2\cos x + 1 - 1 = 0 - 1
By simplifying the above step,
⇒2cos⁡x=−1\Rightarrow 2\cos x = - 1
Now, divide by 2 into both sides.
⇒cos⁡x=−12\Rightarrow \cos x = - \dfrac{1}{2}
Hence, the solutions of the given quadratic equation are sin⁡x=0\sin x = 0 and cos⁡x=−12\cos x = - \dfrac{1}{2}.
Here, the value of sine function is 0 at the angle of 0,π\pi , and 2π2\pi .
And the value of the cosine function is at the angle of ±2π3 \pm \dfrac{{2\pi }}{3}.
This common value that we get with,
x=0,π,2π,2π3,−2π3x = 0,\pi ,2\pi ,\dfrac{{2\pi }}{3}, - \dfrac{{2\pi }}{3}
So the solution set

\Rightarrow S = \left\\{ {0,\pi ,2\pi ,\dfrac{{2\pi }}{3}, - \dfrac{{2\pi }}{3}} \right\\}

Note:
Here, we must remember the trigonometry ratios and the value of ratio at the angles 0, 30, 45, 60, and 90. We also have to learn about the values in all four quadrants with a positive and negative sign.
Some real-life application of trigonometry:

Used to measure the heights of buildings or mountains.
Used in calculus.
Used in physics.
Used in criminology.
Used in marine biology.
Used in cartography.
Used in a satellite system.