Question

Find the value of \(\cos^{-1}\Big(\dfrac12\Big)+2\sin^{-1}\Big(\dfrac12\Big)\).

Hint:

Use the standard values cos^-1(1/2)=pi/3 and sin^-1(1/2)=pi/6.

Answer 1

Step 1: Recognising the reference angles:
\(\tfrac12=\cos(60^\circ)=\sin(30^\circ)\), so \(\cos^{-1}(1/2)=60^\circ=\pi/3\) rad and \(\sin^{-1}(1/2)=30^\circ=\pi/6\) rad.

Step 2: Adding in degrees, then converting:
\(60^\circ+2(30^\circ)=60^\circ+60^\circ=120^\circ=\dfrac{2\pi}{3}\) radians.

Final Answer:
\[ \boxed{\dfrac{2\pi}{3}} \]