Question

How do you find the solution to \[{\tan ^2}\theta - 3 = 0\] if \[0 < \theta < 360\] ?

Answer 1

Hint: The question is related to the trigonometry topic. Here in this question to find the value of \[\theta \] , first we solve the given equation and we use inverse to find the value of \[\theta \] . To find the exact value we use the table of trigonometry ratios for standard angles and hence find the solution for the question.

Complete step-by-step answer:
The sine, cosine, tangent, cosecant, secant and cotangent are the trigonometry ratios of trigonometry. It is abbreviated as sin, cos, tan, cosec, sec and cot. Here in this question, we have \[{\tan ^2}\theta - 3 = 0\] .
Take -3 to the RHS of the equation
 \[ \Rightarrow {\tan ^2}\theta = 3\]
Take square root on the both sides to cancel the square in the LHS of the equation
 \[ \Rightarrow \tan \theta = \pm \sqrt 3 \] ----- (1)
By applying the inverse to the equation we solve the equation.
To find the value we use the table of trigonometry ratios for standard angles.
The table of tangent function for standard angles is given as

Angle	0	30	45	60	90
tan	0	\[\dfrac{1}{{\sqrt 3 }}\]	\[1\]	\[\sqrt 3 \]	1

Since the \[\theta \] lies between 0 and 360 we consider the positive value.
 \[\tan \theta = \sqrt 3 \]
So taking the inverse function we have
 \[ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\sqrt 3 } \right)\]
From the table of tangent function for standard angles
 \[ \Rightarrow x = {60^ \circ }\]
This is in the form of degree; let us convert into radians.
To convert the degree into radian we multiply the degree by \[\dfrac{\pi }{{180}}\]
Therefore, we have \[x = 60 \times \dfrac{\pi }{{180}}\]
On simplification we have
 \[ \Rightarrow x = \dfrac{\pi }{3}\]
Hence, we have solved the given trigonometric function.
Therefore, the value of \[\theta \] is \[\dfrac{\pi }{3}\] in radians and the value of \[\theta \] is \[{60^ \circ }\] in degree.
So, the correct answer is “ \[\dfrac{\pi }{3}\] in radians and the value of \[\theta \] is \[{60^ \circ }\] in degree ”.

Note: The trigonometry and inverse trigonometry are inverse for each other. The inverse of a function is represented as the arc of the function or the function is raised by the power -1. For the trigonometry and the inverse trigonometry, we need to know about the table of trigonometry ratios for the standard angles.