Question

Define trigonometric equations.

Answer 1

Hint: In order to write the answer of this question, we should know the concept of trigonometric equations, that is, we solve trigonometric equations to find out all sets of values of θ. We should also have knowledge of trigonometric ratios to get a more accurate answer.
Complete step-by-step answer:
In this question, we are asked to define trigonometric equations. We know that trigonometric equations are the equations which are solved to get all possible values of θ from the equation. So, we can define trigonometric equations as an equation which involves one or more trigonometric equations. A trigonometric equation can be written as Q1 (sin θ, cos θ, tan θ, cot θ, sec θ, cosec θ) = Q2 (sin θ, cos θ, tan θ, cot θ, sec θ, cosec θ), where Q1 and Q2 are rational functions.
While solving the trigonometric equations, we need to remember the general solutions of a few trigonometric equations like,
If $\sin \theta =0$, then $\theta =n\pi$
If $\cos \theta =0$, then $\theta =n\pi +\dfrac{\pi }{2}$
If $\tan \theta =0$, then $\theta =n\pi$
If $\sin \theta =\sin \alpha$, then $\theta =n\pi +{{\left( -1 \right)}^{n}}\alpha$ where $\alpha \in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$.
If $\cos \theta =\cos \alpha$, then $\theta =2n\pi \pm \alpha$ where $\alpha \in \left[ 0,\pi \right]$.
If $\tan \theta =\tan \alpha$ then $\theta =n\pi +\alpha$ where $\alpha \in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right)$.
Therefore, to use these properties, we try to form an equation in any one ratio.

Note: As we know that trigonometric equations are solved to find the set of all possible values of unknown angles. So, we should have some knowledge of trigonometric ratios. We should also remember that trigonometric equations are not trigonometric identities because they do not satisfy conditions for all angles.