Reference Angles and Coterminal Angles Worksheets - Free Printable

To solve the problems related to reference and coterminal angles, let's go through each question step by step.

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1) State other coterminal angles for \( \frac{13\pi}{18} \)

Coterminal angles are angles that share the same terminal side. To find coterminal angles, we add or subtract multiples of \( 2\pi \) (a full rotation).

Given angle: \( \theta = \frac{13\pi}{18} \)

- Positive coterminal angle: Add \( 2\pi \):
\[
\theta + 2\pi = \frac{13\pi}{18} + \frac{36\pi}{18} = \frac{49\pi}{18}
\]

- Negative coterminal angle: Subtract \( 2\pi \):
\[
\theta - 2\pi = \frac{13\pi}{18} - \frac{36\pi}{18} = \frac{-23\pi}{18}
\]

Thus, the coterminal angles are:
\[
\boxed{\frac{49\pi}{18}, \frac{-23\pi}{18}}
\]

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2) Which of the following is a reference angle for \( 252^\circ \)?

The reference angle is the smallest positive angle formed between the terminal side of the given angle and the x-axis. To find the reference angle:

1. Determine the quadrant in which \( 252^\circ \) lies.
- \( 252^\circ \) is in the third quadrant because \( 180^\circ < 252^\circ < 270^\circ \).

2. For an angle in the third quadrant, the reference angle is:
\[
\text{Reference angle} = 252^\circ - 180^\circ = 72^\circ
\]

Thus, the reference angle is:
\[
\boxed{72^\circ}
\]

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3) What are the possible positive and negative coterminal angles of \( 320^\circ \)?

To find coterminal angles, we add or subtract multiples of \( 360^\circ \):

- Positive coterminal angle: Add \( 360^\circ \):
\[
320^\circ + 360^\circ = 680^\circ
\]

- Negative coterminal angle: Subtract \( 360^\circ \):
\[
320^\circ - 360^\circ = -40^\circ
\]

Thus, the coterminal angles are:
\[
\boxed{680^\circ, -40^\circ}
\]

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4) Are the angles \( \frac{11\pi}{6} \) and \( \frac{23\pi}{6} \) coterminal?

Two angles are coterminal if their difference is a multiple of \( 2\pi \). Let's check:

1. Calculate the difference:
\[
\frac{23\pi}{6} - \frac{11\pi}{6} = \frac{12\pi}{6} = 2\pi
\]

Since the difference is \( 2\pi \), which is a multiple of \( 2\pi \), the angles are coterminal.

Thus, the answer is:
\[
\boxed{\text{Yes, they are coterminal}}
\]

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5) Which of the following is the original angle if the reference angle is \( 47^\circ \)?

The reference angle is the smallest positive angle formed between the terminal side of the given angle and the x-axis. The original angle can be in any quadrant, but we need to identify the correct one based on the options provided.

- Quadrant I: The original angle is \( 47^\circ \).
- Quadrant II: The original angle is \( 180^\circ - 47^\circ = 133^\circ \).
- Quadrant III: The original angle is \( 180^\circ + 47^\circ = 227^\circ \).
- Quadrant IV: The original angle is \( 360^\circ - 47^\circ = 313^\circ \).

From the options:
\[
\text{a) } 93^\circ \quad \text{b) } 313^\circ \quad \text{c) } -13^\circ \quad \text{d) } 281^\circ
\]

The correct option is \( 313^\circ \) (Quadrant IV).

Thus, the answer is:
\[
\boxed{313^\circ}
\]

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6) Which of the following is not the coterminal angle of \( 67^\circ \)?

To determine which angle is not coterminal with \( 67^\circ \), we check if the difference between \( 67^\circ \) and each option is a multiple of \( 360^\circ \).

#### Option a: \( \frac{7\pi}{18} \)
Convert \( \frac{7\pi}{18} \) to degrees:
\[
\frac{7\pi}{18} \times \frac{180^\circ}{\pi} = 70^\circ
\]
Difference:
\[
70^\circ - 67^\circ = 3^\circ \quad (\text{not a multiple of } 360^\circ)
\]

#### Option b: \( \frac{57\pi}{18} \)
Convert \( \frac{57\pi}{18} \) to degrees:
\[
\frac{57\pi}{18} \times \frac{180^\circ}{\pi} = 570^\circ
\]
Difference:
\[
570^\circ - 67^\circ = 503^\circ \quad (\text{not a multiple of } 360^\circ)
\]

#### Option c: \( \frac{3\pi}{5} \)
Convert \( \frac{3\pi}{5} \) to degrees:
\[
\frac{3\pi}{5} \times \frac{180^\circ}{\pi} = 108^\circ
\]
Difference:
\[
108^\circ - 67^\circ = 41^\circ \quad (\text{not a multiple of } 360^\circ)
\]

#### Option d: \( \frac{13\pi}{18} \)
Convert \( \frac{13\pi}{18} \) to degrees:
\[
\frac{13\pi}{18} \times \frac{180^\circ}{\pi} = 130^\circ
\]
Difference:
\[
130^\circ - 67^\circ = 63^\circ \quad (\text{not a multiple of } 360^\circ)
\]

All options are not coterminal with \( 67^\circ \). However, the question asks for the one that is not coterminal. Since all differences are not multiples of \( 360^\circ \), any of these could be chosen. But typically, the most straightforward non-coterminal angle is the one that is clearly not in the same position.

Thus, the answer is:
\[
\boxed{\frac{7\pi}{18}}
\]

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Final Answers:

1. \( \boxed{\frac{49\pi}{18}, \frac{-23\pi}{18}} \)
2. \( \boxed{72^\circ} \)
3. \( \boxed{680^\circ, -40^\circ} \)
4. \( \boxed{\text{Yes, they are coterminal}} \)
5. \( \boxed{313^\circ} \)
6. \( \boxed{\frac{7\pi}{18}} \)