Question

If A, B and C are the interior angles of a right-angle triangle, right-angled at B then find the value of A, given that $\tan 2A=\cot (A–{30}^{\circ })$ and 2A is an acute angle.

Answer 1

Using the trigonometric ratio of complementary angles,

$\cot ({90}^{\circ }-A)=\tan A$

From this ratio, we can write the above expression as:

$\Rightarrow \tan 2A=\cot ({90}^{\circ }-2A)$ ….(1)

Given expression is $\tan 2A=\cot (A–{30}^{\circ })$ ….(2)

Now, equate the equation (1) and (2), we get

$\cot ({90}^{\circ }-2A)=\cot (A–{30}^{\circ })$

$\Rightarrow {90}^{\circ }-2A=A–{30}^{\circ }$

$\Rightarrow 3A={90}^{\circ }+{30}^{\circ }$

$\Rightarrow 3A={120}^{\circ }$

$\Rightarrow A=\frac{{120}^{\circ }}{3}$

$\Rightarrow A={40}^{\circ }$

Thus, the measure of the acute angle A can be easily calculated by making use of the trigonometry ratio of complementary angles.