Question

How do you simplify $\cos (\pi -\theta )$?

Answer 1

Hint:
First convert $\pi$ to a degree that is ${180}^{\circ }$ then apply cosine difference identity to simplify.

Complete step by step solution:
Here , we can apply cosine difference identity to solve this question
We know, $\cos (A-B)=\cos A\cos B+\sin A\sin B$
For this question let’s take $\pi$ as A and $\theta$ as B
Now putting the values in the above formula
$\cos (\pi -\theta )=\cos \pi \cos \theta +\sin \pi \sin \theta$
We know that $\pi ={180}^{\circ }$ , putting the value of $\pi$ in the above equation.
$\Rightarrow \cos ({180}^{\circ }-\theta )=\cos {180}^{\circ }\cos \theta +\sin {180}^{\circ }\sin \theta$
$\cos {180}^{\circ }=(-1)$ and $\sin {180}^{\circ }=0$ , putting the values of cosine and sine in the above equation we get,
$\Rightarrow (-1)\times \cos \theta +0\times \sin \theta$
Multiply and combine the terms
$\Rightarrow -\cos \theta$

Thus, the value of $\cos (\pi -\theta )$ is $-\cos \theta$.

Additional information :
Here in this question $\cos (\pi -\theta )$$=\cos ({180}^{\circ }-\theta )$
If you see the graph , $({180}^{\circ }-\theta )$ lies in the second quadrant and in the second quadrant only sine and cosec is positive which implies that the second quadrant cosine sign will be negative.
Which means you can directly write $\cos (\pi -\theta )$= $-\cos \theta$ by giving appropriate reason.
If you know in which quadrant trigonometric ratios are positive or negative its easier to solve the question and you can also cross check your answer using the graph.

Note:
In the first quadrant all trigonometric ratios are positive. In the second quadrant only sine and its reciprocal cosec is positive rest trigonometric ratios are negative , similarly in third quadrant tan and cot is positive rest are negative and in fourth quadrant cos and sec is positive and rest are negative.
You can memorize this rule by its easy abbreviate that is ASTC , here A stands for all trigonometric ratio, S stands for sine and its reciprocal cosec , T stands for tan and its reciprocal cot, C stand for cosine and its reciprocal sec.