Question

How do you solve ${\tan ^2}x - 3 = 0$ and find all the solutions in the interval $0 \leqslant x < 360$ ?

Answer 1

Hint: Firstly, add the constant $3$ on both sides of the equation. Now, square root on both sides. The square on $\tan x$ and the square root get canceled and on the RHS we will be left with the square root of $3$. Now find the values of $\tan x$ which will be equal to the square root of $3$ and in the given interval $0 \leqslant x < 360$.

Complete step-by-step answer:
The given trigonometric expression is, ${\tan ^2}x - 3 = 0$
Now add the constant $3$ on both sides of the equation.
$\Rightarrow {\tan ^2}x - 3 + 3 = 0 + 3$
On evaluating we get,
$\Rightarrow {\tan ^2}x = 3$
Now apply square root on both sides of the equation.
We do this to cancel out the square on $\tan x$
$\Rightarrow \sqrt {{{\tan }^2}x} = \sqrt 3$
$\Rightarrow \tan x = \pm \sqrt 3$
The values of $\tan x = \sqrt 3$ occurs only when $x = 60^\circ ,240^\circ$ in the given interval, $0 \leqslant x < 360$
And another case which is, $\tan x = - \sqrt 3$ occurs only when $x = 120^\circ ,300^\circ$ in the given interval, $0 \leqslant x < 360$
$\therefore$ The solution of the trigonometric expression, ${\tan ^2}x - 3 = 0$ is when $x = 60^\circ, 120^\circ, 240^\circ, 300^\circ$

Additional information: Whenever complex equations are given to solve one must always Firstly start from the complex side and then convert all the terms into $\cos \theta$ or $\sin \theta$. Then combine them into single fractions. Now it’s most likely to use Trigonometric identities for the transformations if there are any. Know when and where to apply the Subtraction-Addition formula.

Note:
Always check when the trigonometric functions are given in degrees or radians. Whenever we are canceling the square root with a square the solution will always be in the form of $\pm$ . Always check where both the trigonometric functions become negative or positive. Most of the problems can easily be solved by memorizing Quotient identities and Subtraction-Addition identities.