Question

Find the principal and general solutions of the following equations: cosec x = $-2$.

Answer 1

Hint : The principal and general solutions of a given trigonometric ratio refers to the general form of value of given trigonometric ratio. The value of trigonometric ratios is always calculated in a right-angled triangle. There are four quadrants in which the plane is divided.

Complete step-by-step answer :
All six trigonometric ratios are positive in first quadrant, sinθ and cosecθ are positive in second quadrant while other trigonometric ratios are negative, tanθ and cotθ are positive in third quadrant while other trigonometric ratios are negative and cosθ and secθ are positive in fourth quadrant while other trigonometric ratios are negative.
As cosec x = $-2$ is negative, it means that angle x lies in the third quadrant or fourth quadrant in which cosec x is negative. The general solution of angle in the third quadrant is $\pi$ + $\theta$ while in the fourth quadrant; the general solution is 2 $\pi$ - $\theta$ . This makes general solution of cosec x = $-2$ to be equal to $\pi +\dfrac{\pi }{6}=\dfrac{7\pi }{6}$ and $2\pi -\dfrac{\pi }{6}=\dfrac{11\pi }{6}$ .
This makes principal solution of equation, cosec x = $-2$ equal to $n\pi +{{(-1)}^{n}}\dfrac{7\pi }{6}$ such that n belongs to integers (I).
The general form of solution of the equation helps in finding the value of cosec x at other values of x in an easy way.
So, the correct answer is “ $n\pi +{{(-1)}^{n}}\dfrac{7\pi }{6}$ ”.

Note : There are $6$ main trigonometric ratios in all. There are three pairs of trigonometric ratios – (sinθ, cosecθ), (cosθ, secθ) and (tanθ, cotθ). The trigonometric ratio is written as sinθ which means the ratio of perpendicular to Hypotenuse but it does not mean product of sin and $\theta$ .