Question

How do you find the exact value of $2{\sin ^2}\theta - \tan \theta \cot \theta = 0$ in the interval $0 < \theta < 360$?

Answer 1

Hint: In this question, we want to find the trigonometry angle value of the given equation between the intervals 0 to 360. First, we apply the trigonometry formula $\cot \theta = \dfrac{1}{{\tan \theta }}$. After that, $\tan \theta $ is cancelled out from the numerator and the denominator. Then, simplify it and find the value of $\sin \theta $. Based on that value, we will be able to find the value of the angle.

Complete step-by-step answer:
In this question, given that
$ \Rightarrow 2{\sin ^2}\theta - \tan \theta \cot \theta = 0$
As we already know, the cot function is reciprocal of the tan function. That is,
$\cot \theta = \dfrac{1}{{\tan \theta }}$
Let us substitute the values in the given equation.
$ \Rightarrow 2{\sin ^2}\theta - \tan \theta \times \dfrac{1}{{\tan \theta }} = 0$
Now, apply division. So, $\tan \theta $ is cancelled out from the numerator and the denominator. We get,
$ \Rightarrow 2{\sin ^2}\theta - 1 = 0$
Let us add 1 on both sides.
$ \Rightarrow 2{\sin ^2}\theta = 1$
Divide the above step by 2.
$ \Rightarrow {\sin ^2}\theta = \dfrac{1}{2}$
Apply square root on both sides.
$ \Rightarrow \sin \theta = \pm \sqrt {\dfrac{1}{2}} $
Sine value is $\sqrt {\dfrac{1}{2}} $ at the angle of 45 in the first quadrant. In trigonometry angle 45 is also written as$\dfrac{\pi }{4}$. Same way, we can find all the values.
This common value that we get with,
$x = \dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4},\dfrac{{7\pi }}{4},...$
So the solution set

$ \Rightarrow S = \left\{ {x/x = \dfrac{\pi }{4} + \dfrac{k}{2}\pi ,where\ k \in \mathbb{R}} \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.