Question

Find the value of $\sin 765{}^\circ $

Answer 1

Hint: We will find the value of 765 in multiples of 180 and some remainder and with the help of that we will look at in which quadrant it lies and then after that we will see whether sin is positive or negative in that quadrant and find its value using that information.

Complete step-by-step answer:

Let’s start writing the solution.
First we will write 765 in as multiples of 180 and some remainder.
 $765=180\times 4+45$
Now we know that for $180\times 2$degree rotation we cover one full cycle.
So for $180\times 4$ degree we will cover two full cycles and we are at the same point from where we have started.
Now we will again rotate 45 degrees and we will be in the 1st quadrant.
We know that sin is positive in the 1st quadrant.
We know that the value of $\sin 45=\dfrac{1}{\sqrt{2}}$
Now we have shown that $\sin 765=\sin 45$
Hence, from this we get,
$\sin 765=\dfrac{1}{\sqrt{2}}$
Hence, we have found the value of $\sin 765$

Note: One can also convert the value of degree into radian by multiplying the value in degree with $\dfrac{\pi }{180}$ and then we know that $2\pi $is for the full rotation of the cycle. And then we will again get the final angle as $\dfrac{\pi }{4}$ which is 45 in degree, and hence we will get the same answer. But the first method involves less calculation so that one should be followed.