# In $$\triangle$$ ABC right angled at B, it tan A = $$1\over \sqrt{3}$$, find the value of (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

## Solution :

Consider a $$\triangle$$ ABC, in which $$\angle$$ B = 90

For $$\angle$$ A, we have :

Base = AB, Perp. = BC,  and   Hyp. = AC,

tan A = $$\perp\over base$$ = $$BC\over AB$$ = $$1\over \sqrt{3}$$

Let BC = k and AB = $$\sqrt{3} k, AC = \(\sqrt{{AB}^2 + {BC}^2}$$ = 2k

$$\therefore$$   sin A = $$\perp\over hyp.$$ = $$BC\over AC$$ = $$k\over 2k$$ = $$1\over 2$$

and,  cos A = $$base\over hyp.$$ = $$AB\over AC$$ = $$\sqrt{3}k\over 2k$$ = $$\sqrt{3}\over 2$$

For $$\angle$$ C, we have :

Base = BC, Perp = AB and Hyp. = AC,

$$\therefore$$   sin C = $$\perp\over hyp.$$ = $$AB\over AC$$ = $$\sqrt{3}k\over 2k$$ = $$\sqrt{3}\over 2$$

and,  cos C = $$base\over hyp.$$ = $$BC\over AC$$ = $$k\over 2k$$ = $$1\over 2$$

(i)  sin A cos C + cos A sin C = 1

(ii)  cos A cos C – sin A sin C = 0