Question

Find the value of $\cos 540^\circ = $
A) \[ - \dfrac{1}{{\sqrt 2 }}\]
B) $\dfrac{1}{{\sqrt 2 }}$
C) $\dfrac{3}{{\sqrt 2 }}$
D) $ - 1$

Answer 1

Hint: Here, we are required to find the value of $\cos 540^\circ $. We will use quadrants to answer this question. We will break the given angle in two angles such that we know the value of either of the angles or that angle is present on the quadrants. After breaking the angle, we will solve them using quadrants and hence, find the required answer.

Complete step by step solution:
$\cos 540^\circ $ can be written as:
$\cos 540^\circ = \cos \left( {360^\circ + 180^\circ } \right)$
Now, we will draw the quadrants:

Now, $\cos 360^\circ $means one complete rotation.
Since, $\cos 360^\circ $ lies nearly on the fourth quadrant and on the fourth quadrant $\cos \theta $ is positive.
Therefore, $\cos 360^\circ $ is also positive.
Now, when we add any angle to $\cos 360^\circ $ it means that we are starting the second rotation. Hence, we usually neglect $360^\circ $ while solving the trigonometric questions.
Since,
$\cos 540^\circ = \cos \left( {360^\circ + 180^\circ } \right)$
Now, this can be written as:
$\cos 540^\circ = \cos 180^\circ $
Now, we can find the value of $\cos 180^\circ $in two ways:
$\cos 180^\circ = \cos \left( {180^\circ - 0^\circ } \right)$
Now, when we subtract any angle from $180^\circ $, then, we reach the second quadrant.
In the second quadrant, cosine is negative and due to $180^\circ $, the trigonometric function will remain the same.
$ \Rightarrow \cos \left( {180^\circ - 0^\circ } \right) = - \cos 0^\circ $
As we know, $\cos 0^\circ = 1$
Hence, $\cos 180^\circ = - \cos 0^\circ = - 1$
$\cos 180^\circ = \cos \left( {270^\circ - 90^\circ } \right)$
Now, when we subtract any angle from $270^\circ $, then, we reach the third quadrant.
In the third quadrant, cosine is negative and also, due to $270^\circ $, the trigonometric function will change, i.e. cosine will change to sine.
$ \Rightarrow \cos \left( {270^\circ - 90^\circ } \right) = - \sin 90^\circ $
As we know, $\sin 90^\circ = 1$
Hence, $\cos \left( {270^\circ - 90^\circ } \right) = - \sin 90^\circ = - 1$
Therefore, the value of $\cos 180^\circ $ is $ - 1$.
Hence, $\cos 540^\circ = \cos 180^\circ = - 1$
Therefore, the required answer is $ - 1$

And, option D is the correct answer.

Note:
We should take care while solving the quadrants because in one quadrant cosine is positive and in the other it is negative. If we use $0^\circ ,180^\circ ,360^\circ $then the trigonometric functions remain the same. But if we use $90^\circ ,270^\circ $then, we have changed them. Also, we can break the given angle in any sum possible but we should keep in mind that we have to break it in such a sum such that we know the value of one of those or how to solve the question using one of those angles.