Question

Prove that $\cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ = 0$ .

Answer 1

Hint: In this question, we can see that we have trigonometric expressions. So here we will apply the trigonometric identities or formulas to solve this question. We can see that we have all the values in the cosine function with different angles. So we will try to analyze in which quadrant the following values of the cosine function fall and then according to that we will simplify them.

Complete step-by-step answer:
Here we have the expression $\cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ $ .
If we try to remember we can understand that $\cos 306^\circ $ falls in the fourth quadrant and the value of cosine in this quadrant is positive.
Now we have another term $\cos 234^\circ $ . We can see that this angel falls in the third quadrant and this quadrant the value of cosine function is negative.
Similarly, for $\cos 162^\circ $ , it lies in the second quadrant, and in this quadrant the sign of $\cos $ is negative.
For the last term, we have $\cos 18^\circ $ . It will lie in the first quadrant and the value of the cosine function is always positive.
We can write
$306 = (360 - 306)$ .
It gives us $54^\circ $ .
Similarly, for the second angle, we can write $234$ as
 $234 - 180 = 54$
For the third angle, we have $162^\circ $ .
We can write this as
$180^\circ - 162^\circ = 18^\circ $
And for the last angle, we will leave the value as it is.
Now we will substitute these values in the equation and also we will write the sign of the function according to their quadrant value as we have checked.
So we have :
$\cos 54^\circ + ( - \cos 54^\circ ) + ( - \cos 18^\circ ) + \cos 18^\circ $
Upon simplifying we have
$\cos 54^\circ - \cos 54^\circ - \cos 18^\circ + \cos 18^\circ $
Since all the terms get canceled so It gives us value $0$
Hence it is proved that $\cos 306^\circ + \cos 234^\circ + \cos 162^\circ + \cos 18^\circ = 0$ .

Note: Before solving these types of questions we should always be aware of the basic trigonometric identities and functions. We should know that in the first quadrant all the trigonometric ratios are positive. In the second quadrant only sine and cosecant are positive, in the third quadrant only $\tan ,\cot $ are positive and in the fourth quadrant only $\cos ,\sec $ are positive and the rest are negative.
One of the basic identity of the cosine function is:
$\cos (180^\circ - \theta ) = - \cos \theta $
$\cos (180^\circ + \theta ) = - \cos \theta $
$\cos (360^\circ - \theta ) = \cos \theta $