Question

If the lengths of the sides of a right-angled triangle $ABC,$ right angled at $C,$ are in arithmetic progression, then the value of $5\left(sin A + sin B\right)$ is

Answer 1

$\left(a + d\right)^{2}=\left(a - d\right)^{2}+a^{2}$ (Pythagoras theorem)
$\cancel{a^{2}}+\cancel{d^{2}}+2ad=\cancel{a^{2}}+\cancel{d^{2}}-2ad+a^{2}\Rightarrow 4ad=a^{2}\Rightarrow a=4d$
$sinA=\frac{a - d}{a + d}=\frac{4 d - d}{4 d + d}=\frac{3}{5}$
$sinB=\frac{a}{a + d}=\frac{4 d}{5 d}=\frac{4}{5}$
$\Rightarrow 5sinA+5sinB=3+4=7$
$\Rightarrow 5\left(sin A + sin B\right)=7$