Question

In ΔABC, right angled at B. If tan A = `1/sqrt3` , find the value of

(i) sin A cos C + cos A sin C

(ii) cos A cos C − sin A sin C

Answer 1

 

`tan A = 1/sqrt3`

`(BC)/(AB) = 1/ sqrt3`

If BC is k, then AB will be `sqrt3k` , where k is a positive integer.

In ΔABC,

AC² = AB² + BC²

`= (sqrt3k)^2 + (k)^2`

= 3k² + k² = 4k²

∴ AC = 2k

`sin A = ("Side adjacent to"angleA)/"Hypotenuse" = (BC)/(AC) = k/(2k) = 1/2`

`cos A = ("Side adjacent to"angleA)/"Hypotenuse" = (AB)/(AC) = (sqrt3k)/(2k) = sqrt3/2`

`sin C = ("Side adjacent to"angleC)/"Hypotenuse" = (AB)/(AC) = (sqrt3k)/(2k) = sqrt3/2`

`cos C = ("Side adjacent to"angleC)/"Hypotenuse" = (BC)/(AC) = (k)/(2k) = 1/2`

(i) sin A cos C + cos A sin C

`= (1/2)(1/2)+(sqrt3/2)(sqrt3/2) =  1/4 ++ 3/4`

= 4/4 = 1

(ii) cos A cos C − sin A sin C

`= (sqrt3/2)(1/2)-(1/2)(sqrt3/2) =  sqrt3/4 - sqrt3/4 = 0`