Trignometric-equation Formula Sheet

Trigonometric equations are mathematical expressions that involve trigonometric functions (like sine, cosine, and tangent) set equal to a value, and solving them typically requires finding the angles that satisfy the equation within given intervals.

Neetesh Kumar | May 19, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Solution of Trigonometric Equation
• 3. General Solution of Trigonometric Equations

1. Definition:

It is an equation involving one or more trigonometrical ratios of unknown angles.

2. Solution of Trigonometric Equation:

A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation.
$\bold{Principal \space Solution:}$ The solution of the trigonometric equation lying in the interval [$0, 2\pi$].
$\bold{General \space Solution:}$ Since all the trigonometric functions are many one & periodic, hence there are infinite values of $\theta$ for which trigonometric functions have the same value. A general formula gives all possible values of $\theta$ for which the given trigonometric function is satisfied. Such a general formula is called the general solution of a trigonometric equation.

3. General Solution of Trigonometric Equations:

$\bold{(a)}$ If Sin$\theta$ = 0, then $\theta = n\pi, n \in I$
$\bold{(b)}$ If cos$\theta$ = 0, then $\theta = (2n+1)\frac{\pi}{2}, n \in I$
$\bold{(c)}$ If tan$\theta$ = 0, then $\theta = n\pi, n \in I$
$\bold{(d)}$ If Sin$\theta$ = Sin$\alpha$, then $\theta = n\pi + (-1)^n \alpha, n \in I$
$\bold{(e)}$ If Cos$\theta$ = Cos$\alpha$, then $\theta = 2n\pi \plusmn \alpha, n \in I$
$\bold{(f)}$ If tan$\theta$ = tan$\alpha$, then $\theta = n\pi \plusmn \alpha, n \in I$
$\bold{(g)}$ If Sin$\theta$ = 1, then $\theta = 2n\pi + \frac{\pi}{2}, (4n+1)\frac{\pi}{2}, n\in I$
$\bold{(h)}$ If Cos$\theta$ = 1, then $\theta = 2n\pi, n\in I$
$\bold{(i)}$ If Sin$^2\theta$ = Sin$^2\alpha$ or Cos$^2\theta$ = Cos$^2\alpha$ or tan$^2\theta$ = tan$^2\alpha,$ then $\theta = n\pi \plusmn \alpha, n\in I$
$\bold{(j)}$ Sin$(n\pi + \theta) = (-1)^n$Sin$\theta, n \in I$
$\bold{(k)}$ Cos$(n\pi + \theta) = (-1)^n$Cos$\theta, n \in I$

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