Question

In triangle ABC, right-angled at B, if tan A =\(\frac{ 1}{\sqrt{3}}\). 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 = \(\frac{1}{\sqrt3}\)

\(\frac{BC}{AB} = \frac{1}{\sqrt3}\)

Given BC = k, then AB = \(\sqrt3 k\), where k is a positive integer.
In \(Δ\) ABC,
\(\text{AC}^ 2 =\text{ AB}^ 2 + \text{BC}^ 2\)
\(\text{AC}^ 2 =(\sqrt3 k)^2+(k)^2\)
AC² = 3k² + k²
AC² = 4k²
AC = 2k

\(\text{Sin A} =\frac{ BC}{AC} =\frac{ 1}{2}\)

\(\text{Cos A} =\frac{ AB}{AC} = \frac{\sqrt3}{2}\)

\(\text{Sin C} =\frac{ AB}{AC} = \frac{\sqrt3}{2}\)

\(\text{Cos C} = \frac{BC}{AC} = \frac{1}{2}\)

(i) sin A cos C + cos A sin C = \(\left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) + \frac{\sqrt3}{2} \times \frac{\sqrt3}{2} = \frac{1}{4} + \frac{3}{4} = 1\)

(ii) cos A cos C – sin A sin C = \(\left(\frac{\sqrt3}{2}\right)\left(\frac{1}{2}\right) – \left(\frac{1}{2}\right) \left(\frac{\sqrt3}{2}\right) = 0\)