# Thread: how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

1. ## how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

hi I NEED help on my A Level maths i am confused on ,
how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3)/6)

really appreciate it if you could help

Sunset

2. Originally Posted by sunset
hi I NEED help on my A Level maths i am confused on ,
how to prove arcsin(0.5)+arcsine(1/3) = arcsin((2(2^0.5)+3^0.5)/6)

really appreciate it if you could help

Sunset
let $\displaystyle a = \arcsin\left(\frac{1}{2}\right)$

$\displaystyle \sin{a} = \frac{1}{2}$ ... $\displaystyle \cos{a} = \frac{\sqrt{3}}{2}$

$\displaystyle b = \arcsin\left(\frac{1}{3}\right)$

$\displaystyle \sin{b} = \frac{1}{3}$ ... $\displaystyle \cos{b} = \frac{2\sqrt{2}}{3}$

$\displaystyle \sin(a+b) = \sin{a}\cos{b} + \cos{a}\sin{b}$

$\displaystyle \sin(a+b) = \frac{1}{2} \cdot \frac{2\sqrt{2}}{3} + \frac{\sqrt{3}}{2} \cdot \frac{1}{3}$

$\displaystyle \sin(a+b) = \frac{2\sqrt{2} + \sqrt{3}}{6}$

$\displaystyle a+b = \arcsin\left(\frac{2\sqrt{2} + \sqrt{3}}{6}\right)$