Solveeit Logo
Login

Question

Question: How do you find the exact value of \(2{\sin ^2}\theta - \tan \theta \cot \theta = 0\) in the interva...

How do you find the exact value of 2sin⁡2θ−tan⁡θcot⁡θ=02{\sin ^2}\theta - \tan \theta \cot \theta = 0 in the interval 0<θ<3600 < \theta < 360?

Explanation

Solution

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⁡θ=1tan⁡θ\cot \theta = \dfrac{1}{{\tan \theta }}. After that, tan⁡θ\tan \theta is cancelled out from the numerator and the denominator. Then, simplify it and find the value of sin⁡θ\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
⇒2sin⁡2θ−tan⁡θcot⁡θ=0\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⁡θ=1tan⁡θ\cot \theta = \dfrac{1}{{\tan \theta }}
Let us substitute the values in the given equation.
⇒2sin⁡2θ−tan⁡θ×1tan⁡θ=0\Rightarrow 2{\sin ^2}\theta - \tan \theta \times \dfrac{1}{{\tan \theta }} = 0
Now, apply division. So, tan⁡θ\tan \theta is cancelled out from the numerator and the denominator. We get,
⇒2sin⁡2θ−1=0\Rightarrow 2{\sin ^2}\theta - 1 = 0
Let us add 1 on both sides.
⇒2sin⁡2θ=1\Rightarrow 2{\sin ^2}\theta = 1
Divide the above step by 2.
⇒sin⁡2θ=12\Rightarrow {\sin ^2}\theta = \dfrac{1}{2}
Apply square root on both sides.
⇒sin⁡θ=±12\Rightarrow \sin \theta = \pm \sqrt {\dfrac{1}{2}}
Sine value is 12\sqrt {\dfrac{1}{2}} at the angle of 45 in the first quadrant. In trigonometry angle 45 is also written asπ4\dfrac{\pi }{4}. Same way, we can find all the values.
This common value that we get with,
x=π4,3π4,5π4,7π4,...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.