Find the reference angle for each of the following: 210 degrees, 135 degrees, 5pi/3 and 5pi/4.

Step-by-step solution

1. 1

   Know what a reference angle is. The reference angle of $\theta$ is the acute angle between the terminal side of $\theta$ and the $x$-axis — never the $y$-axis. It is always in $(0^\circ,90^\circ)$, i.e. $\left(0,\tfrac{\pi}{2}\right)$ in radians, and it is what lets you read the size of $\sin\theta$, $\cos\theta$ or $\tan\theta$ off the first quadrant and then attach the correct sign.

2. 2

   Learn the one rule per quadrant. For $\theta$ between $0$ and one full turn:

   Quadrant	Degrees	Radians
   I	$\theta$	$\theta$
   II	$180^\circ-\theta$	$\pi-\theta$
   III	$\theta-180^\circ$	$\theta-\pi$
   IV	$360^\circ-\theta$	$2\pi-\theta$

   Each line simply measures back to whichever half of the $x$-axis is nearer. If $\theta$ is negative or larger than a full turn, add or subtract $360^\circ$ ($2\pi$) first.

3. 3

   a. $210^\circ$. Since $180^\circ<210^\circ<270^\circ$, the angle is in Quadrant III, so subtract $180^\circ$:

   $210^\circ-180^\circ=30^\circ$

4. 4

   b. $135^\circ$. Since $90^\circ<135^\circ<180^\circ$, the angle is in Quadrant II, so subtract it from $180^\circ$:

   $180^\circ-135^\circ=45^\circ$

5. 5

   c. $\dfrac{5\pi}{3}$. Write the boundaries with the same denominator: $\pi=\tfrac{3\pi}{3}$, $\tfrac{3\pi}{2}=\tfrac{4.5\pi}{3}$ and $2\pi=\tfrac{6\pi}{3}$. Since $\tfrac{5\pi}{3}$ lies between $\tfrac{4.5\pi}{3}$ and $\tfrac{6\pi}{3}$, it is in Quadrant IV:

   $2\pi-\frac{5\pi}{3}=\frac{6\pi}{3}-\frac{5\pi}{3}=\frac{\pi}{3}$

6. 6

   d. $\dfrac{5\pi}{4}$. With $\pi=\tfrac{4\pi}{4}$ and $\tfrac{3\pi}{2}=\tfrac{6\pi}{4}$, the angle $\tfrac{5\pi}{4}$ sits in Quadrant III:

   $\frac{5\pi}{4}-\pi=\frac{5\pi}{4}-\frac{4\pi}{4}=\frac{\pi}{4}$

7. 7

   Check every answer against the sine values. A reference angle must reproduce the magnitude of the trig ratios: $\sin 210^\circ=-\tfrac12$ and $\sin 30^\circ=\tfrac12$; $\sin 135^\circ=\tfrac{\sqrt2}{2}$ and $\sin 45^\circ=\tfrac{\sqrt2}{2}$; $\sin\tfrac{5\pi}{3}=-\tfrac{\sqrt3}{2}$ and $\sin\tfrac{\pi}{3}=\tfrac{\sqrt3}{2}$; $\sin\tfrac{5\pi}{4}=-\tfrac{\sqrt2}{2}$ and $\sin\tfrac{\pi}{4}=\tfrac{\sqrt2}{2}$. In each case only the sign differs, exactly as it should. Note that $210^\circ$ and $\tfrac{5\pi}{4}$ are both Quadrant III yet give different reference angles — the quadrant chooses the rule, not the answer.