Question

If tan x = `1(1)/(3)`, find the value of : 4 sin²x - 3 cos²x + 2

Answer 1

Consider the diagram below :

tan x = `1(1)/(3)`

tan x = `(4)/(3)`

i.e.`"perpendicular"/"base" = (4)/(3)`

Therefore if length of base = 3x, length of perpendicular = 4x

Since

base²+ perpendicular² = hypotenuse²  ..[ Using Pythagoras Theorem]

(3x)² + (4x)² = hypotenuse²

hypotenuse = 9x²+16x²+ 25x²

∴ hypotenuse = 5x

Now

sin x = `"perpendicular"/"hypotenuse" = (4x)/(5x) = (4)/(5)`

cos x = `"base"/"hypotenuse" = (3x)/(5x) = (3)/(5)`

Therefore
4 sin² x – 3cos² x +2

= `4(4/5)^2 – 3(3/5)^2+ 2`

= `(64)/(25) – (27)/(25)+ 2`

= `(87)/(25)`

=`3(12)/(25)`