Question

How do you evaluate $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)?$

Answer 1

First we have to see that where is $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$ in the quadrant. So, we can go graphing $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$.Otherwise we can take the help of the co-terminals concept and get the points and then solve it.

Complete step by step answer:
According to the question, we will first check in which quadrant does $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$ lies. We will graph $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$. If the angle is positive then, it will start in an anti-clockwise direction, else it will go in clockwise direction.As we can see that the angle is negative, so we will start from in clockwise direction. We always start from the x-axis. When we point $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$ in the graph, we get that $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$ comes under the 4th Quadrant.

Our point will be $\dfrac{\pi }{4}$ and the coordinates are $\left( {\dfrac{{\sqrt 2 }}{2}, - \dfrac{{\sqrt 2 }}{2}} \right)$. If we notice then we can see that we have completed one revolution, so we need to always find out how far it goes after completing one round. So, we will write:
$( - 2\pi + t)$
Now, $2\pi$ in terms of the denominator $4$ is:
$\sin \left( {\dfrac{{ - 8\pi }}{4} + t} \right)$
Now, according to question, we find that:
$\Rightarrow \sin \left( {\dfrac{{ - 9\pi }}{4}} \right) = \sin \left( {\dfrac{{ - 8\pi }}{4} - \dfrac{\pi }{4}} \right)$
When we see that we can write it as a period name, then we simply write it as:
$\Rightarrow \sin \left( {\dfrac{{ - 9\pi }}{4}} \right) = \sin \left( { - \dfrac{\pi }{4}} \right)$
This makes it a lot simpler and easier.
When we see the coordinates, we can tell that:
$\therefore \sin \left( { - \dfrac{\pi }{4}} \right) = \left( { - \dfrac{{\sqrt 2 }}{2}} \right)$
This is negative here because the y-axis is negative.

Note: To point $\sin \left( {\dfrac{{ - 9\pi }}{4}} \right)$ in the graph, we have to see how many $\pi$ over $4$ are there. According to the question, we know that it is $9\pi$ over $4$. So, we can divide the ‘x’ and ‘y’ axis into $4$ parts, where each axis is getting $4$ parts. After that we can calculate where $9\pi$ over $4$ lies, and then we can get our quadrant.