cos(x-pi) Simplify | Value of cos(pi-x)

The value of cos(x-pi) is equal to -cosx. This can be done using the trigonometric formulas of compound angles. In this post, we will learn how to simplify cos(pi-x) and cos(x-pi).

We will use the following formula on compound angles to find cos(pi-x) and cos(x-pi).

$\cos(a-b)=\cos a \cos b +\sin a \sin b$ $\cdots (\star)$

Value of cos(x-pi)

Question: Simplify $\cos(x-\pi)$.

Answer:

In the above formula $(\star)$ we put $a=x$ and $b=\pi$. Then we obtain that

$\cos(x-\pi)$ $=\cos x \cos \pi +\sin x \sin \pi$

$=\cos x \cdot (-1)+\sin x \cdot 0$ as we know that $\cos(\pi)=-1$ and $\sin(\pi) = 0$.

$=-\cos x +0$

$=-\cos x$

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

Value of cos(pi-x)

Question: Simplify $\cos(\pi-x)$.

Answer:

In the above formula $(\star)$ we put $a=\pi$ and $b=x$. Then we obtain that

$\cos(\pi-x)$ $=\cos \pi \cos x + \sin \pi \sin x$

$=\cos x \cdot (-1)+\sin x \cdot 0$ as $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

$=-\cos x +0$ $=-\cos x$

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

Also Read:

sinx=0, cosx=0, tanx=0 General Solution

Values of sin 15, cos 15, tan 15

Values of sin 75, cos 75, tan 75

Question 1: Find the value of $\cos \dfrac{3\pi}{4}$.

Answer:

From the above, we know that $\cos(\pi-x)=-\cos x$. Here, we put $x=\pi/4$ to get the value of $\cos \frac{3\pi}{4}$. By doing so we obtain that

$\cos(\pi-\pi/4)=-\cos \pi/4$

$\Rightarrow \cos \frac{3\pi}{4}=-\cos \pi/4$

$\Rightarrow \cos \frac{3\pi}{4}=-\dfrac{1}{2}$

So the value of cos(3pi/4) is equal to -1/2.

FAQs