Question

If \[tan{\text{ }}x{\text{ }} = 3cotx\] find $x$ in radian ?

Answer 1

Hint: We can solve this question by applying the trigonometric formula of , where $x$ are angles. In the fourth quadrant the tangent and cotangent remains positive along with the first quadrant where all the wanted sine, cosine, tangent remains positive. Also remember the trigonometry table so that we can quickly assign the values as per the angles , inversely in this question we will find the angle by checking the value 3 as given.

Complete step-by-step solution:
We are given the question of finding the x if \[tan{\text{ }}x{\text{ }} = 3cotx\] . We are going to solve it by applying the trigonometric values corresponding to the angles . And the property of reciprocal as follows -
As we know that there is property in trigonometry that the cotangent is the reciprocal of the tangent . \[{\text{cot }}x{\text{ }} = \dfrac{1}{{\tan x}}\] .
In order to solve \[tan{\text{ }}x{\text{ }} = 3cotx\]
\[tan{\text{ }}x{\text{ }} = 3 \times cotx\]
We can apply the reciprocal property … \[{\text{cot }}x{\text{ }} = \dfrac{1}{{\tan x}}\]
\[tan{\text{ }}x{\text{ }} = 3 \times \dfrac{1}{{\tan x}}\]
By multiplying the \[tan{\text{ }}x{\text{ }}\]both the sides L. H. S. and R. H. S. of the ‘ equal to ‘ sign .
We get ,
\[ta{n^2}x{\text{ }} = 3\]
Here , we need to take square root ,
\[tanx{\text{ }} = \sqrt 3 \]
Now , we will apply inverse trigonometry or in other words we can say that we can determine the angle when the corresponding value is given to us as –
\[x{\text{ }} = {\tan ^{ - 1}}\left( {\sqrt 3 } \right)\]
The value \[\sqrt 3 \] comes when the tangent is having the angle of \[{60^ \circ }\].
Therefore , the angel comes to be \[{60^ \circ }\]and if we want to convert in the radian then ,
\[radian = \dfrac{\pi }{{180}} \times degree\]
Substitute the value of the degree in the formula , we get –
\[radian = \dfrac{\pi }{{180}} \times {60^ \circ }\]
\[radian = \dfrac{\pi }{{180}} \times {60^ \circ }\]
= $\dfrac{\pi }{3}$radian
Putting value of $\pi = 3.14$,
Radian = $1.0472$

Therefore , the answer to this question in radian is $\dfrac{\pi }{3}$ rad or $1.0472$ rad or \[{60^ \circ }\] in degree.

Additional Information : In Mathematics the inverse trigonometric functions (every so often additionally called anti-trigonometric functions or cyclomatic function) are the reverse elements of the mathematical functions In particular, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are utilized to get a point from any of the point's mathematical proportions.

Even Function – A function $f(x)$ is said to be an even function ,if $f( - x) = f(x)$for all x in its domain.
Odd Function – A function $f(x)$ is said to be an even function ,if $f( - x) = - f(x)$for all x in its domain.
We know that $\sin ( - \theta ) = - \sin \theta .\cos ( - \theta ) = \cos \theta \,and\,\tan ( - \theta ) = - \tan \theta $.

Note: Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus .
i) In inverse trigonometric function, the domain are the ranges of corresponding trigonometric functions and the range are the domain of the corresponding trigonometric function .
ii) As a matter of fact, and and their reciprocals, and are odd functions whereas and its reciprocal are even functions .
iii) SI unit of measure of any angle is radian .
iv) One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer .