Question

How do you find the values of $\theta$ given $\tan \theta = 1$?

Answer 1

The question is involving the trigonometric function. The tangent is one of the trigonometry ratios. Here in this question, by taking inverse tangent function on both sides of a given function then using the specified angle of trigonometric ratios we can get the required value of angle $\theta$.

Complete step by step explanation:
Tangent or tan is the one of the trigonometric function defined as the ratio between the opposite
side and adjacent side of right angled triangle with the angle $\theta$

The value of specified angles of the tan function is
$\tan {0^ \circ } = 0$
$\tan {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$
$\tan {45^ \circ } = 1$
$\tan {60^ \circ } = \sqrt 3$
$\tan {90^ \circ } = \infty$

Now, Consider the given equation
$\Rightarrow \,\,\,\tan \theta = 1$

Take inverse function of tan i.e., ${\tan ^{ - 1}}$ on both side, then
$\Rightarrow \,\,{\tan ^{ - 1}}\left( {\,\tan \theta } \right) = {\tan ^{ - 1}}\left( 1 \right)$

As we now the $x.{x^{ - 1}} = 1$ , then
$\Rightarrow \,\,1.\theta = {\tan ^{ - 1}}\left( 1 \right)$
$\Rightarrow \,\,\theta = {\tan ^{ - 1}}\left( 1 \right)$

By the value specified angle
$\Rightarrow \,\,\theta = {45^ \circ }$

Now we convert angle$\theta$ degree to radian by multiplying $\dfrac{\pi }{{180}}$, then
$\Rightarrow \,\,\theta = 45 \times \dfrac{\pi }{{180}}$
$\therefore \,\,\,\,\,\theta = {\dfrac{\pi }{4}^c}$

However, the tangent function is positive in the first and third quadrants. To find the
second solution, add the reference angle from $\pi$ to find the solution in the fourth quadrant.
$\Rightarrow \,\,\theta = \pi + \dfrac{\pi }{4}$
By taking 4 has LCM in RHS
$\Rightarrow \,\,\theta = \dfrac{{4\pi + \pi }}{4}$
$\therefore \,\,\,\,\,\theta = {\dfrac{{5\pi }}{4}^c}$

The period of the $\tan (\theta )$ function is $\pi$ so value 1 will repeat every $\pi$ radians in
both directions.
$\Rightarrow \,\,\,\,\theta = \dfrac{\pi }{4} + n\pi ,\,\,\,\dfrac{{5\pi }}{4} + n\pi$, for any
integer$n$

In general ,
$\therefore \,\,\,\,\theta = \dfrac{\pi }{4} + n\pi$, for any integer $n$

Hence, the values of $\theta$ given$\tan \theta = 1$ is $\theta = \dfrac{\pi }{4} + n\pi$, for any integer $n$

Note: The trigonometry and inverse trigonometry is inverse of each other. To solve this kind of problem we need trigonometry and inverse trigonometry concepts. To find the value of $\theta$ We use the inverse trigonometry concept. We have a table for trigonometry ratios for standard angles, using this we determine the value of $\theta$. And we can also verify the answer by substituting the value.