SOLUTION: Coterminal angles and reference angles - Studypool - Free Printable

We are given six angles (in degrees and radians), some negative, some large, and we're asked to find the reference angle for each. A reference angle is the smallest positive acute angle between the terminal side of the given angle and the x-axis. It's always between $0^\circ$ and $90^\circ$ (or $0$ and $\frac{\pi}{2}$ radians).

Let’s go through each problem step by step.

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1) $-230^\circ$

Step 1: Find a coterminal angle between $0^\circ$ and $360^\circ$.

Since the angle is negative, we add $360^\circ$:
$$
-230^\circ + 360^\circ = 130^\circ
$$

So, $-230^\circ$ is coterminal with $130^\circ$, which lies in Quadrant II.

Step 2: Reference angle in Quadrant II:
$$
\text{Reference angle} = 180^\circ - 130^\circ = 50^\circ
$$

✔ Answer: $50^\circ$

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2) $-\frac{25\pi}{18}$

Step 1: Convert to a positive coterminal angle.

Add $2\pi$ (since $2\pi = \frac{36\pi}{18}$):
$$
-\frac{25\pi}{18} + \frac{36\pi}{18} = \frac{11\pi}{18}
$$

Now, $\frac{11\pi}{18}$ is between $0$ and $2\pi$. Let's determine the quadrant:

- $\frac{\pi}{2} = \frac{9\pi}{18}$, so $\frac{11\pi}{18} > \frac{9\pi}{18}$ → Quadrant II.

Step 2: Reference angle in Quadrant II:
$$
\text{Reference angle} = \pi - \frac{11\pi}{18} = \frac{18\pi}{18} - \frac{11\pi}{18} = \frac{7\pi}{18}
$$

✔ Answer: $\frac{7\pi}{18}$

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3) $-\frac{7\pi}{9}$

Step 1: Add $2\pi = \frac{18\pi}{9}$ to get a positive coterminal angle:
$$
-\frac{7\pi}{9} + \frac{18\pi}{9} = \frac{11\pi}{9}
$$

Now, $\frac{11\pi}{9}$ is between $\pi = \frac{9\pi}{9}$ and $\frac{3\pi}{2} = \frac{13.5\pi}{9}$, so it's in Quadrant III.

Step 2: Reference angle in Quadrant III:
$$
\text{Reference angle} = \frac{11\pi}{9} - \pi = \frac{11\pi}{9} - \frac{9\pi}{9} = \frac{2\pi}{9}
$$

✔ Answer: $\frac{2\pi}{9}$

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4) $-\frac{29\pi}{18}$

Step 1: Add $2\pi = \frac{36\pi}{18}$:
$$
-\frac{29\pi}{18} + \frac{36\pi}{18} = \frac{7\pi}{18}
$$

$\frac{7\pi}{18}$ is less than $\frac{\pi}{2} = \frac{9\pi}{18}$, so it's in Quadrant I.

In Quadrant I, the reference angle is the angle itself.

✔ Answer: $\frac{7\pi}{18}$

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5) $\frac{31\pi}{9}$

This is greater than $2\pi$, so reduce it modulo $2\pi$.

First, compute how many full rotations:
$$
2\pi = \frac{18\pi}{9}, \quad \frac{31\pi}{9} \div \frac{18\pi}{9} = \frac{31}{18} \approx 1.722
$$

Subtract $2\pi = \frac{18\pi}{9}$ once:
$$
\frac{31\pi}{9} - \frac{18\pi}{9} = \frac{13\pi}{9}
$$

Now, $\frac{13\pi}{9}$ is still greater than $\pi = \frac{9\pi}{9}$, and less than $\frac{3\pi}{2} = \frac{13.5\pi}{9}$, so it's in Quadrant III.

Step 2: Reference angle in Quadrant III:
$$
\frac{13\pi}{9} - \pi = \frac{13\pi}{9} - \frac{9\pi}{9} = \frac{4\pi}{9}
$$

✔ Answer: $\frac{4\pi}{9}$

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6) $640^\circ$

Step 1: Reduce by subtracting $360^\circ$:
$$
640^\circ - 360^\circ = 280^\circ
$$

Still greater than $360^\circ$? No — $280^\circ < 360^\circ$, so it's the coterminal angle.

Now, $280^\circ$ is between $270^\circ$ and $360^\circ$, so it's in Quadrant IV.

Step 2: Reference angle in Quadrant IV:
$$
360^\circ - 280^\circ = 80^\circ
$$

✔ Answer: $80^\circ$

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✔ Final Answers:

1) $50^\circ$
2) $\frac{7\pi}{18}$
3) $\frac{2\pi}{9}$
4) $\frac{7\pi}{18}$
5) $\frac{4\pi}{9}$
6) $80^\circ$

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🔍 Summary of Steps:

To find the reference angle:
1. Find a coterminal angle between $0^\circ$ and $360^\circ$ (or $0$ and $2\pi$).
2. Determine the quadrant.
3. Use the appropriate formula:
- QI: same as angle
- QII: $180^\circ - \theta$
- QIII: $\theta - 180^\circ$
- QIV: $360^\circ - \theta$

For radians:
- QI: same
- QII: $\pi - \theta$
- QIII: $\theta - \pi$
- QIV: $2\pi - \theta$

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