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Question: If two angles of a triangle are \({\tan ^{ - 1}}(2)\) and \({\tan ^{ - 1}}\left( 3 \right)\) , then ...

If two angles of a triangle are tan⁡−1(2){\tan ^{ - 1}}(2) and tan⁡−1(3){\tan ^{ - 1}}\left( 3 \right) , then the third angle is
A. π4\dfrac{\pi }{4}
B. π6\dfrac{\pi }{6}
C. π3\dfrac{\pi }{3}
D.π2\dfrac{\pi }{2}

Explanation

Solution

Hint: Here we will proceed by using the property of tan inverse. Then by applying the conditions given in the question we will get our required answer.

Complete step-by-step answer:
The inverse trigonometric functions of sine, cosecant, secant, and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics etc. But most of the time, the convention symbol represents the inverse trigonometric functions using arc-prefix like arcsin(x), arctan(x), arccsc(x), arcsec(x), arccot(x).
Consider the function y = f(x), and x = g(y) then the inverse of the function is written as g=f−1^{ - 1}
Given two angles are tan⁡−1(2){\tan ^{ - 1}}\left( 2 \right) and tan⁡−1(3){\tan ^{ - 1}}\left( 3 \right). Now, (2)(3)>1\left( 2 \right)\left( 3 \right) > 1
⇒tan⁡−1(x)+tan⁡−1(y) =π+tan⁡−1x+y1−x×y ⇒tan⁡−1(2)+tan⁡−1(3) =π+tan⁡−12+31−2×3 ⇒π+tan⁡−1(−1) =π−π4 =3π4  \Rightarrow {\tan ^{ - 1}}\left( x \right) + {\tan ^{ - 1}}\left( y \right) \\\ = \pi + {\tan ^{ - 1}}\dfrac{{x + y}}{{1 - x \times y}} \\\ \Rightarrow {\tan ^{ - 1}}\left( 2 \right) + {\tan ^{ - 1}}\left( 3 \right) \\\ = \pi + {\tan ^{ - 1}}\dfrac{{2 + 3}}{{1 - 2 \times 3}} \\\ \Rightarrow \pi + {\tan ^{ - 1}}\left( { - 1} \right) \\\ = \pi - \dfrac{\pi }{4} \\\ = \dfrac{{3\pi }}{4} \\\
Therefore, the third angle is
π−3π4 =π4  \pi - \dfrac{{3\pi }}{4} \\\ = \dfrac{\pi }{4} \\\
Hence, the A is the correct option.

Note: Whenever we come up with this type of problem, it should be noted that to solve the different types of trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. By taking the reference of inverse trigonometric functions one can easily solve this question.