Question

Find the value of $\cos 24^\circ+\cos 55^\circ+\cos 125^\circ+\cos 156^\circ$.

• A. $0$
• B. $2$
• C. $3$
• D. $1$

Hint:

Whenever you see angles that are supplements ($\theta$ and $180^\circ-\theta$), pair them using $\cos(180^\circ-\theta)=-\cos\theta$ to simplify quickly.

Answer 1

The Correct Option is A

Step 1: Use $\cos(180^\circ-\theta)=-\cos\theta$.
$\cos 156^\circ=\cos(180^\circ-24^\circ)=-\cos 24^\circ$, \quad $\cos 125^\circ=\cos(180^\circ-55^\circ)=-\cos 55^\circ$. Step 2: Pair and sum.
\[ (\cos24^\circ+\cos156^\circ)+(\cos55^\circ+\cos125^\circ) =(\cos24^\circ-\cos24^\circ)+(\cos55^\circ-\cos55^\circ)=0. \] \[ \boxed{0} \]