Question

thanks

Answer 1

sin^-1(1/2) is the angle whose sine is 1/2 and that is 30 degrees

tan-1(sqrt(3)/3) is also 30 degrees
The third side of the triangle is sqrt(3^2 + (sqrt(3))^2) = sqrt(12) = 2*sqrt(3)
Divide all sides by sqrt(3) and we have a triangle with sides sqrt(3), 1 and 2

cos(30 + 30) = cos(60) = 1/2

Answer 2

\[\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b) \\ \text{ where } \\ a=\arctan(\frac{\sqrt{3}}{3}) \text{ and } b=\arcsin(\frac{1}{2}) \\ \]
draw two right triangles for both a and b

Answer 3

thanks!!!