Question

Define principal 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 those values which lie between 0 degree to 360 degree is our principal 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 $\pi$. If the equation involves a variable $0\le x<2\pi$, then the solutions are called principal solutions.
Example- Consider the equation $\sin \theta =\dfrac{1}{2}$. This equation is, clearly, satisfied by $\theta$ equal to $\dfrac{\pi }{6}$ and $\dfrac{5\pi }{6}$. So these are its solutions. Solving an equation means to find the set of all values of the unknown value which satisfy the given equation. The solutions lying between 0 to $2\pi$ or between ${{0}^{\circ }}$ to ${{360}^{\circ }}$ are called principal solutions. Clearly we see that principal solution of the equation $\sin \theta =\dfrac{1}{2}$ are $\dfrac{\pi }{6}$ and $\dfrac{5\pi }{6}$ because these solutions lie between 0 to $2\pi$.

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 principal solution.