Question

If \[90^\circ < A < 180^\circ\], and \[\sin A = \dfrac{4}{5}\]. Then what is the value of \[\tan\left( {\dfrac{A}{2}} \right)\]?
A. \[\dfrac{1}{2}\]
B. \[\dfrac{3}{5}\]
C. \[\dfrac{3}{2}\]
D. 2

Answer 1

Hint: First calculate the range of the half angle. Then use trigonometric identity \[\sin^{2}x + \cos^{2}x = 1\] and calculate the value of \[\cos A\]. In the end, substitute the values of \[\sin A\] and \[\cos A\]in the half angle formula of tangent function to reach the required answer.

Formula used:
1. \[\sin^{2}x + \cos^{2}x = 1\]
2. The half-angle formula: \[\tan\left( {\dfrac{x}{2}} \right) = \dfrac{{\sin x}}{{1 + \cos x}} = \dfrac{{1 - \cos x}}{{\sin x}}\]

Complete step by step solution:
Given: \[90^\circ < A < 180^\circ\] and \[\sin A = \dfrac{4}{5}\]
Let’s calculate the range of the half angle.
\[90^\circ < A < 180^\circ\]
Divide each term by 2.
\[45^\circ < \dfrac{A}{2} < 90^\circ\]
Now apply the trigonometric identity \[\sin^{2}x + \cos^{2}x = 1\].
\[\sin^{2}A + \cos^{2}A = 1\]
Substitute \[\sin A = \dfrac{4}{5}\] in the above equation.
\[{\left( {\dfrac{4}{5}} \right)^2} + \cos^{2}A = 1\]
\[ \Rightarrow \]\[\dfrac{{16}}{{25}} + \cos^{2}A = 1\]
\[ \Rightarrow \]\[\cos^{2}A = 1 - \dfrac{{16}}{{25}}\]
\[ \Rightarrow \]\[\cos^{2}A = \dfrac{9}{{25}}\]
Take square root on both sides.
\[\cos A = \pm \dfrac{3}{5}\]
since \[A\] lies in the second quadrant. And the value of trigonometric function \[\cos\] in the second quadrant is negative.
Hence, \[\cos A = - \dfrac{3}{5}\]
Now apply the half angle formula \[\tan\left( {\dfrac{x}{2}} \right) = \dfrac{{\sin x}}{{1 + \cos x}}\].
\[\tan\left( {\dfrac{A}{2}} \right) = \dfrac{{\sin A}}{{1 + \cos A}}\]
Substitute \[\cos A = - \dfrac{3}{5}\] and \[\sin A = \dfrac{4}{5}\] in the above equation.
\[\tan\left( {\dfrac{A}{2}} \right) = \dfrac{{\dfrac{4}{5}}}{{1 + \left( { - \dfrac{3}{5}} \right)}}\]
Simplify the equation.
\[\tan\left( {\dfrac{A}{2}} \right) = \dfrac{{\dfrac{4}{5}}}{{\dfrac{2}{5}}}\]
\[ \Rightarrow \]\[\tan\left( {\dfrac{A}{2}} \right) = \dfrac{4}{2}\]
\[ \Rightarrow \]\[\tan\left( {\dfrac{A}{2}} \right) = 2\]
Hence the correct option is D.

Note: There is another way of finding the value of \[\tan\left( {\dfrac{A}{2}} \right)\]. Use the trigonometric formula \[\sin\left( {2x} \right) = \dfrac{{2\tan x}}{{1 + \tan^{2}x}}\] for the given trigonometric equation \[\sin A = \dfrac{4}{5}\] and simplify it. The simplified equation is a quadratic equation. So, factorize the equation and calculate the value of the required angle.