Question

Q.15) If cos A =2/5, find the value of 4 + 4 tan^2A

Answer 1

Hey there!

Given,

cos A = 2 / 5

base / hypotenuse = 2 / 5

or, AB / AC = 2/5

Let AB = 2x and AC = 5x.

Now,

By pythagoras theorem,

$BC = \sqrt{AC^2 - AB^2}\\BC = \sqrt{(5x)^2 - (2x)^2}\\BC = \sqrt{25x^2 - 2x^2}\\BC = \sqrt{21x^2}\\BC = \sqrt{21}x$

Now,

tan A = Perpendicular / base

tan A = BC / AB

tan A = $\dfrac{\sqrt{21}x}{2x} = \dfrac{\sqrt{21}}{2}$

So,

4 + 4 tan² A

$= 4 + 4 \left(\dfrac{\sqrt{21}}{2}\right)^2\\\\= 4 + 4 \times \dfrac{21}{4}\\\\=4 + 21\\=25$