Question

Find the value of $\cos {540}^{\circ }=$
A) $$-{\displaystyle \frac{1}{\sqrt{2}}}$$
B) ${\displaystyle \frac{1}{\sqrt{2}}}$
C) ${\displaystyle \frac{3}{\sqrt{2}}}$
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.