Question

In a right-angled triangle, it is given that A is an acute angle and tan A = `(5) /(12)`.

find the value of :
(i) cos A
(ii) sin A
(iii)  ` (cosA+sinA)/(cosA– sin A)`

Answer 1

Consider the diagram below : 

tan A = `(5)/(12)`

i.e.`"perpendicular"/"base" = (5)/(12) ⇒ "BC"/"AB" = (5)/(12)`

Therefore if length of AB = 12x, length of BC = 5x

Since
AB² + BC² = AC²     ...[ Using Pythagoras Theorem ]

(12x)² + (5x)² = AC²

AC² = 144x² + 25x²

AC² = 169x²

∴ AC = 13x                    ...( hypotenuse)

(i) cos A = `"base"/"hypotenuse" = "AB"/"AC" = (12x)/(13x) = 12/13`

(ii) sin A = `"perpendicular"/"hypotenuse" = (5x)/(13x) = 5/13`

(iii) `(cos "A" + sin "A")/(cos "A" – sin "A")`

= `(12/13+5/13)/(12/13 – 5/13)` 

= `(17/13)/(17/7)`

= `17/7`

= `2(3)/(7)`