Question

How do you find the solution to ${{\tan }^{2}}\theta +4\tan \theta -12=0$ if $0\le \theta \le 360?$

Answer 1

Hint: We will add and subtract necessary terms to and from the given trigonometric equation. Then we will take the greatest common factors out. Then, we will find out the factors of the equation. From that, we will be able to find the solution.

Complete step by step solution:
Let us consider the given trigonometric identity ${{\tan }^{2}}\theta +4\tan \theta -12=0.$
We will add and subtract $2\tan \theta $ on the left-hand side of the given equation.
We will get ${{\tan }^{2}}\theta +6\tan \theta -2\tan \theta -12=0.$
Let us take the common term$\tan \theta $ from the first term and the second term out and take the common term $-2$ from the third term and the fourth term.
Then, we will get the equation as $\tan \theta \left( \tan \theta +6 \right)-2\left( \tan \theta +6 \right)=0.$
As we can see, there is still a common term. The term is $\tan \theta +6.$ Let us take this term out.
We will get the equation in the form of $\left( \tan \theta +6 \right)\left( \tan \theta -2 \right)=0.$
So, from what we have learnt earlier, we can say $\tan \theta +6=0$ or $\tan \theta -2=0.$
Let us consider the term $\tan \theta +6.$ We will get $\tan \theta +6=0.$
From this equation, we will get $\tan \theta =-6.$
And from this step, we can find the value of $\theta $ for which $\tan \theta +6=0.$
We can find it as $\theta ={{\tan }^{-1}}-6=-80.54{}^\circ =360-80.54=279.46{}^\circ .$
Also, we will get $\theta =279.46-180=99.46{}^\circ .$
This is because the Tangent function is negative in the second quadrant and the fourth quadrant.
Let us consider the other term $\tan \theta -2.$ It will give us $\tan \theta -2=0.$
This will lead us to obtain $\tan \theta =2.$
Thus, we can find the value of $\theta $ for which $\tan \theta -2=0$ by $\theta ={{\tan }^{-1}}2=63.43{}^\circ .$
Also, we will get $\theta =180+63.43=243.43{}^\circ .$
This is because the Tangent function is positive in the first quadrant and the second quadrant.

Hence the solution of the given equation is $0\le \theta \le 360, \theta =63.43{}^\circ ,99.46{}^\circ ,243.43{}^\circ ,279.46{}^\circ .$

Note: We can solve this equation by putting $\tan \theta =x.$ The equation will become ${{x}^{2}}+4x-12=0.$ This can be factorized to get $\left( x-2 \right)\left( x+6 \right)=0.$ So, we will get $x=2$ or $x=-6.$ Therefore, $\tan \theta =2$ or $\tan \theta =-6.$