Question

The most general values of $\theta$ satisfying $tan{\text{ }}\theta {\text{ }} + {\text{ }}tan{\text{ }}\left( {\dfrac{{3\pi }}{4}{\text{ }} + {\text{ }}\theta } \right){\text{ }} = {\text{ }}2$are given by
\left( 1 \right)$$$$2n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{3},{\text{ }}n \in Z
\left( 2 \right)$$$$n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{3},{\text{ }}n \in Z
\left( 3 \right)$$$$2n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{6},{\text{ }}n \in Z
\left( 4 \right)$$$$n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{6},{\text{ }}n \in Z

Answer 1

Hint : We have to find the general value of $\theta$ . We solve this by using the trigonometric identities and the general values of the trigonometric functions . We also know the tan function is the ratio of sin function to cos function . Using the trigonometric identities of tan and cot functions and general solutions of trigonometric functions . On simplifying the equation we can find the value of $\theta$.

Complete step-by-step answer :
Given :
$tan{\text{ }}\theta {\text{ }} + {\text{ }}tan{\text{ }}\left( {\dfrac{{3\pi }}{4}{\text{ }} + {\text{ }}\theta } \right){\text{ }} = {\text{ }}2$
Also , $\dfrac{{3\pi }}{4}$can be written as ${\text{ }}\dfrac{\pi }{2} + \dfrac{\pi }{4}$
$tan{\text{ }}\theta {\text{ }} + {\text{ }}tan\left[ {\dfrac{\pi }{2}{\text{ }} + {\text{ }}\left( {\dfrac{\pi }{4}{\text{ }} + {\text{ }}\theta } \right)} \right]{\text{ }} = {\text{ }}2$
We know , $tan{\text{ }}\left[ {{\text{ }}\dfrac{\pi }{2}{\text{ }} + {\text{ }}\theta {\text{ }}} \right]{\text{ }} = {\text{ }} - {\text{ }}cot{\text{ }}\theta$
$tan{\text{ }}\theta {\text{ }}-{\text{ }}cot{\text{ }}\left( {\dfrac{\pi }{4}{\text{ }} + {\text{ }}\theta } \right){\text{ }} = {\text{ }}2$———(1)
Using the formula of $cot{\text{ }}\left( {x{\text{ }} + {\text{ }}y} \right){\text{ }} = {\text{ }}\dfrac{{\left( {cot{\text{ }}x{\text{ }} \times {\text{ }}cot{\text{ }}y{\text{ }} - {\text{ }}1} \right)}}{{\left( {cot{\text{ }}x{\text{ }} + {\text{ }}cot{\text{ }}y} \right)}}$
Applying in equation $\left( 1 \right)$
$tan{\text{ }}\theta {\text{ }}-{\text{ }}\left[ {{\text{ }}\dfrac{{\left( {cot{\text{ (}}\dfrac{\pi }{4}{\text{) }} \times {\text{ }}cot{\text{ }}\theta {\text{ }}-{\text{ }}1} \right)}}{{\left( {cot{\text{ (}}\dfrac{\pi }{4}{\text{) }} + {\text{ }}cot{\text{ }}\theta } \right){\text{ }}}}} \right]{\text{ }} = {\text{ }}2$
As , $cot{\text{ }}\dfrac{\pi }{4}{\text{ }} = {\text{ }}1$
$tan{\text{ }}\theta {\text{ }}-{\text{ }}\left[ {{\text{ }}\dfrac{{\left( {cot{\text{ }}\theta {\text{ }}-{\text{ }}1} \right)}}{{\left( {1{\text{ }} + {\text{ }}cot{\text{ }}\theta } \right)}}{\text{ }}} \right]{\text{ }} = {\text{ }}2$
Also, $tan{\text{ }}\theta {\text{ }} = {\text{ }}\dfrac{1}{{cot{\text{ }}\theta }}$
$tan{\text{ }}\theta {\text{ }}-{\text{ }}\left[ {{\text{ }}\dfrac{{\left( {1{\text{ }}-{\text{ }}tan{\text{ }}\theta } \right)}}{{\left( {1{\text{ }} + {\text{ }}tan{\text{ }}\theta } \right)}}{\text{ }}} \right]{\text{ }} = {\text{ }}2$
Simplifying the equation , we get
$tan{\text{ }}\theta {\text{ }}\left( {1{\text{ }} + {\text{ }}tan{\text{ }}\theta } \right){\text{ }}-{\text{ }}\left( {1{\text{ }}-{\text{ }}tan{\text{ }}\theta } \right){\text{ }} = {\text{ }}2{\text{ }}\left( {1{\text{ }} + {\text{ }}tan{\text{ }}\theta } \right)$
On further solving
$tan\theta + ta{n^2}\theta - 1 + tan\theta - 2 - 2tan\theta = 0$
$ta{n^2}\theta = 3$
Taking square root , we get
$tan{\text{ }}\theta {\text{ }} = {\text{ }} \pm \surd 3$
Also , we know value of $tan{\text{ }}\dfrac{\pi }{3}{\text{ }} = {\text{ }} \pm \surd 3$
$tan{\text{ }}\theta {\text{ }} = {\text{ }} \pm {\text{ }}tan{\text{ }}\dfrac{\pi }{3}$
As the general value of $tan{\text{ }}\theta$lies between $\left( {\dfrac{{ - \pi }}{2},{\text{ }}\dfrac{\pi }{2}} \right)$
General equation of $tan{\text{ }}\theta$:
$tan{\text{ }}\theta {\text{ }} = {\text{ }}tan{\text{ }}\alpha$, then$\;\theta {\text{ }} = {\text{ }}n\pi {\text{ }} + {\text{ }}\alpha$, where $n \in Z$
$\theta {\text{ }} = {\text{ }}n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{3},{\text{ }}n \in Z$
hence, the general value of $\theta {\text{ }} = {\text{ }}n\pi {\text{ }} \pm {\text{ }}\dfrac{\pi }{3}{\text{ }},{\text{ }}n \in Z$
Thus the correct option is $\left( 2 \right)$
So, the correct answer is “Option B”.

Note: Equations involving trigonometric functions of a variable are called trigonometric equations . The solutions of a trigonometric equation for $0 \leqslant {\text{ }}x{\text{ }} < {\text{ }}2\pi$( $x$ is the angle of the trigonometric function ) are called principle solutions . The expressions involving integers ${\text{' }}n{\text{ '}}$which give all solutions of a trigonometric equation are called general solutions .
Various general formulas of trigonometric functions :
$sin{\text{ }}\theta {\text{ }} = {\text{ }}sin{\text{ }}\alpha$, then$\theta = n\pi {\text{ }} \pm {( - 1)^n}\alpha$, where $n \in Z$
$cos{\text{ }}\theta {\text{ }} = {\text{ }}cos{\text{ }}\alpha$, then$\theta {\text{ }} = {\text{ }}2n\pi {\text{ }} \pm {\text{ }}\alpha$, where $n \in Z$
$tan{\text{ }}\theta {\text{ }} = {\text{ }}tan{\text{ }}\alpha$, then$\theta {\text{ }} = {\text{ }}n\pi {\text{ }} + {\text{ }}\alpha$, where $n \in Z$