Question

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

Answer 1

Hint: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.