Question

Define general solution of trigonometric equation.

Answer 1

Hint: Trigonometric equations involve functions like sin, cos, tan, cosec, sec and cot. After solving these equations we will get some values and all these values which involves integer n is our general solution.

Complete step-by-step answer:
Before proceeding with the solution we should understand the concept of trigonometric equations. The equations which involve trigonometric functions like sin, cos, tan, cot, sec etc. are called trigonometric equations.
We already know that the values of sin x and cos x repeat after an interval of $2\pi$. Also, the values of tan x repeat after an interval of $\pi$. A general solution is one which involves the integer ‘n’ and gives all solutions of a trigonometric equation. Also, the character ‘Z’ is used to denote the set of integers. Thus, a solution generalized by means of periodicity is known as the general solution.
Example- Consider the equation $\cos \theta =-\dfrac{1}{2}$. This equation is, clearly, satisfied by $\theta$ equal to $\dfrac{2\pi }{3}$ and $\dfrac{4\pi }{3}$, etc. Since the trigonometric functions are periodic, therefore, if a trigonometric equation has a solution, it will have an infinite number of solutions. These solutions can be put together in compact form as $2n\pi \pm \dfrac{2\pi }{3}$ where n is an integer. This solution is known as the general solution.

Note: In solving trigonometric equations we need to remember the formulas, the standard values of angles and the identities because then it becomes easy. We in a hurry can make a mistake in applying the cofunction identities as we can write cos in place of sin and sin in place of cos while solving for the general solution.