Question

How do you find the reference angle for $- \dfrac{{3\pi }}{4}$ ?

Answer 1

Hint : To find the reference angle of a negative angle, we have to first convert it to a positive angle by adding it to the complete angle $(360^\circ )$ , then we will identify the quadrant in which the angle lies. Each quadrant is of 90 degrees, so if an angle lies in the second quadrant then we subtract 90 degrees from it. If the angle lies in the third quadrant then we subtract 180 degrees from it, and if the angle lies in the fourth quadrant then we subtract 270 degrees from it. Thus, after converting the given negative angle to a positive angle, we will identify the quadrant in which the angle will lie and then using the above information, find its reference angle.

Complete step by step solution:
We have to find the reference angle for $- \dfrac{{3\pi }}{4}$
We know that $180^\circ = \pi$
So, the positive value of the given negative angle is
$2\pi - \dfrac{{3\pi }}{4} = \dfrac{{5\pi }}{4}$
This angle is greater than 180 degrees but smaller than 270 degrees, so it lies in the third quadrant.
To find its reference angle, we will subtract 180 degrees or $\pi$ from it –
$\dfrac{{5\pi }}{4} - \pi = \dfrac{\pi }{4}$
Hence the reference angle for $- \dfrac{{3\pi }}{4}$ is $\dfrac{\pi }{4}$ .
So, the correct answer is “ $- \dfrac{{3\pi }}{4}$ is $\dfrac{\pi }{4}$ ”.

Note : An angle whose terminal side is in the second, third or fourth quadrant is known as reference angle. A reference angle is always smaller than 90 degrees, that is, it is an acute angle. The initial side of a reference angle is on the positive x-axis and the terminal side is in the first quadrant. So to find the reference angle of any angle, we have to first identify the quadrant in which the angle lies and then we can find the reference angle as shown above.