Question

How do you sketch the angle $\dfrac{5\pi }{4}$ and find the reference angle?

Answer 1

Hint: Now to find the angle we will first determine in which quadrant does the angle lie in. Now once we have determined we will find the reference angle accordingly. Now to sketch the angle we will use the reference angle and draw an angle in the required measurement. Hence we have the sketch of the above angle.

Complete step by step solution:
Now let us first understand the angles.
The given angle is in radians.
Now first we know that the angles in the interval $\left( 0,\dfrac{\pi }{2} \right)$ lies in the first quadrant. The angles between $\left( \dfrac{\pi }{2},\pi \right)$ lies in the second quadrant. The angles $\left( \pi ,\dfrac{3\pi }{2} \right)$ lies in third quadrant and lastly The angles from $\left( \dfrac{3\pi }{2},2\pi \right)$ lies in fourth quadrant.
Now consider the angle $\dfrac{5\pi }{4}$ .
Now the angle lies in $\left( \dfrac{4\pi }{4},\dfrac{6\pi }{4} \right)$ which is nothing but $\left( \pi ,\dfrac{3\pi }{4} \right)$ .
Hence we can say that the given angle lies in the third quadrant.
Hence now to find reference angle we will consider $\dfrac{5\pi }{4}-\pi $
Now we will make the denominators of the two terms the same by taking LCM of denominators.
LCM of 1 and 4 is 4. Hence we get,
$\begin{align}
  & \Rightarrow \dfrac{5\pi }{4}-\dfrac{4\pi }{4} \\
 & \Rightarrow \dfrac{5\pi -4\pi }{4} \\
 & \Rightarrow \dfrac{\pi }{4} \\
\end{align}$
Hence the reference angle is $\dfrac{\pi }{4}$ .
Now to plot the angle we will draw an angle counterclockwise of $\dfrac{\pi }{4}$ in third quadrant
Hence we get the graph as,

Note: Now note that all the angels mentioned in the problem and the solutions are in radians. To convert the angles from radians to degrees we will multiply the angle by $\dfrac{180}{\pi }$ . Similarly to convert the angles to radians from degrees we will multiply the angles to $\dfrac{\pi }{180}$ . Hence note that $\dfrac{5\pi }{4}={{225}^{\circ }}$ .