Reference angles worksheet with exercises to calculate reference angles for various angles in degrees and radians, and to convert between degrees and radians.
Worksheet titled "Reference Angles" with problems to find reference angles for given angles in degrees and radians, including both positive and negative values, and a section to convert between degrees and radians.
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Step-by-step solution for: Solved Name Date Period Worksheet -Reference Angles Find the ...
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Step-by-step solution for: Solved Name Date Period Worksheet -Reference Angles Find the ...
Let's solve each problem step by step. We'll find reference angles, and convert between degrees and radians as needed.
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The reference angle is the smallest angle formed between the terminal side of the given angle and the x-axis. It is always positive and less than 90° (or π/2 radians).
- In Quadrant I: reference angle = θ
- In Quadrant II: reference angle = 180° − θ
- In Quadrant III: reference angle = θ − 180°
- In Quadrant IV: reference angle = 360° − θ
For negative angles, add 360° until you get a positive equivalent.
For angles > 360°, subtract 360° until it’s within 0° to 360°.
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## ✔ Part 1: Find the reference angle for each angle
- 330° is in Quadrant IV
- Reference angle = 360° − 330° = 30°
✔ Answer: 30°
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- Negative angle → add 360° to make it positive:
- -75° + 360° = 285°
- 285° is in Quadrant IV
- Reference angle = 360° − 285° = 75°
✔ Answer: 75°
---
- 450° > 360° → subtract 360°:
- 450° − 360° = 90°
- 90° lies on the positive y-axis, which is the boundary between Quadrants I and II.
- But since it's exactly 90°, its reference angle is 0°? Wait — actually, 90° is a special case.
But reference angle is defined as the acute angle to the x-axis. For 90°, the angle to the x-axis is 90°, but that's not acute.
Wait — reference angles are measured from the x-axis to the terminal side, so:
- At 90°, the terminal side is on the y-axis → distance to x-axis is 90°, but we usually say the reference angle is 90° only if we allow it?
Actually, standard definition: The reference angle is always between 0° and 90°, inclusive.
So for 90°, the reference angle is 90°? No — wait!
Let’s clarify:
> A reference angle is the acute angle between the terminal side and the x-axis.
At 90°, the terminal side is perpendicular to the x-axis → the angle between them is 90°, but that’s not acute.
However, by convention, when the angle is 90°, 180°, 270°, or 360°, the reference angle is taken as the acute angle formed with the axis, so:
- For 90°, reference angle = 90°? No — actually, many sources define reference angle for quadrantal angles as 0° or undefined, but typically:
> Reference angle for 90° is 90°, but since it's not acute, some say it's undefined.
But more commonly, we consider reference angle for angles on axes as the smallest angle to the axis, so:
- 90° → reference angle = 90° (but not acute), but most textbooks say:
> Reference angle is the acute angle between the terminal side and the x-axis.
So at 90°, the angle between terminal side and x-axis is 90°, which is not acute, so perhaps we say the reference angle is 90°?
Wait — let’s check standard rule:
> The reference angle is the acute angle between the terminal side and the x-axis.
So if the angle is 90°, the angle between terminal side and x-axis is 90°, which is not acute, so does it have a reference angle?
No — but in practice, for quadrantal angles like 90°, 180°, etc., we often say the reference angle is 0° or just leave it as 90°, but most sources say:
> The reference angle for 90° is 90°, even though it's not acute.
But strictly speaking, reference angles are always between 0° and 90°, inclusive.
So 90° is acceptable.
But let’s go with standard textbook logic:
> For 90°, the reference angle is 90°.
But that contradicts “acute”.
Actually, many sources define reference angle as the smallest positive angle between the terminal side and the x-axis, regardless of being acute.
So:
- 90° → reference angle = 90°
But others say it's 0°?
No — better to follow this:
> Reference angle = |θ - 180°n| where n is chosen so that result is in [0°, 90°]
For 90°:
- Closest x-axis: it's on the y-axis → angle to x-axis is 90° → so reference angle = 90°
But 90° is not acute, so some sources say no reference angle.
But in practice, for quadrantal angles, we do assign reference angles:
- 90° → ref = 90°
- 180° → ref = 0°
- 270° → ref = 90°
- 360° → ref = 0°
Wait — actually, standard rule:
> For angles on axes, the reference angle is the angle between the terminal side and the nearest x-axis, so:
- 90° → to x-axis: 90° → ref = 90°
- 180° → to x-axis: 0° → ref = 0°
- 270° → to x-axis: 90° → ref = 90°
- 360° → to x-axis: 0° → ref = 0°
But again, 90° is not acute, so maybe not considered a "reference angle".
But commonly accepted:
> Reference angle for 90° is 90°
But let’s double-check with examples.
Actually, most math sources say:
> The reference angle is the acute angle formed by the terminal side and the x-axis.
So if the terminal side is on the axis, then the angle is 90° or 0°, but 90° is not acute → so no acute angle, so reference angle is not defined?
But that can’t be — we need a consistent method.
Let’s use this method:
> Reduce angle to 0° to 360°, then:
- QI: θ
- QII: 180° − θ
- QIII: θ − 180°
- QIV: 360° − θ
And for quadrantal angles, the reference angle is the smallest angle to the x-axis, which is:
- 90° → 90°
- 180° → 0°
- 270° → 90°
- 360° → 0°
But since 90° is not acute, some say reference angle is undefined, but in practice, we accept 90°.
But let’s look at 450°:
- 450° − 360° = 90°
- So 450° is coterminal with 90°
- 90° is on the positive y-axis → reference angle = 90°
But since 90° is not acute, and reference angles are usually acute, we may say reference angle is 90°, but it’s a special case.
So for 450°:
- Coterminal angle: 90°
- Reference angle: 90°
✔ Answer: 90°
---
- 35° is in Quadrant I
- Reference angle = 35°
✔ Answer: 35°
---
Convert to degrees:
$ \frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ $
- 150° is in Quadrant II
- Reference angle = 180° − 150° = 30°
✔ Answer: 30°
---
Convert to degrees:
$ -\frac{\pi}{2} \times \frac{180^\circ}{\pi} = -90^\circ $
Add 360°:
-90° + 360° = 270°
- 270° is on the negative y-axis
- Reference angle = angle to x-axis = 90°
(From 270° to x-axis is 90°)
✔ Answer: 90°
---
Convert to degrees:
$ \frac{13\pi}{6} \times \frac{180^\circ}{\pi} = \frac{13 \times 180}{6} = 390^\circ $
Reduce: 390° − 360° = 30°
- 30° is in Quadrant I
- Reference angle = 30°
✔ Answer: 30°
---
Convert to degrees:
$ \frac{7\pi}{4} \times \frac{180^\circ}{\pi} = \frac{7 \times 180}{4} = 315^\circ $
- 315° is in Quadrant IV
- Reference angle = 360° − 315° = 45°
✔ Answer: 45°
---
## ✔ Part 2: Find the reference angle for each angle
- 30° is in Quadrant I
- Reference angle = 30°
✔ Answer: 30°
---
- Add 360°: -150° + 360° = 210°
- 210° is in Quadrant III
- Reference angle = 210° − 180° = 30°
✔ Answer: 30°
---
Convert to degrees:
$ \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = \frac{7 \times 180}{6} = 210^\circ $
- 210° is in Quadrant III
- Reference angle = 210° − 180° = 30°
✔ Answer: 30°
---
Convert to degrees:
$ -\frac{5\pi}{2} \times \frac{180^\circ}{\pi} = -\frac{5 \times 180}{2} = -450^\circ $
Now add 360° repeatedly:
- -450° + 360° = -90°
- -90° + 360° = 270°
- 270° is on the negative y-axis
- Reference angle = angle to x-axis = 90°
✔ Answer: 90°
---
## ✔ Part 3: Convert from radians to degrees or degrees to radians
Use: $ \text{radians} = \text{degrees} \times \frac{\pi}{180} $
$ 45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} $
✔ Answer: $ \frac{\pi}{4} $
---
$ -150^\circ = -150 \times \frac{\pi}{180} = -\frac{150\pi}{180} = -\frac{5\pi}{6} $
✔ Answer: $ -\frac{5\pi}{6} $
---
## ✔ Final Answers Summary
1. 330° → 30°
2. -75° → 75°
3. 450° → 90°
4. 35° → 35°
5. $ \frac{5\pi}{6} $ → 30°
6. $ -\frac{\pi}{2} $ → 90°
7. $ \frac{13\pi}{6} $ → 30°
8. $ \frac{7\pi}{4} $ → 45°
13. 30° → 30°
14. -150° → 30°
15. $ \frac{7\pi}{6} $ → 30°
16. $ -\frac{5\pi}{2} $ → 90°
17. 45° = $ \frac{\pi}{4} $
18. -150° = $ -\frac{5\pi}{6} $
---
Let me know if you'd like this formatted as a printable answer sheet!
---
🔷 What is a Reference Angle?
The reference angle is the smallest angle formed between the terminal side of the given angle and the x-axis. It is always positive and less than 90° (or π/2 radians).
- In Quadrant I: reference angle = θ
- In Quadrant II: reference angle = 180° − θ
- In Quadrant III: reference angle = θ − 180°
- In Quadrant IV: reference angle = 360° − θ
For negative angles, add 360° until you get a positive equivalent.
For angles > 360°, subtract 360° until it’s within 0° to 360°.
---
## ✔ Part 1: Find the reference angle for each angle
1. 330°
- 330° is in Quadrant IV
- Reference angle = 360° − 330° = 30°
✔ Answer: 30°
---
2. -75°
- Negative angle → add 360° to make it positive:
- -75° + 360° = 285°
- 285° is in Quadrant IV
- Reference angle = 360° − 285° = 75°
✔ Answer: 75°
---
3. 450°
- 450° > 360° → subtract 360°:
- 450° − 360° = 90°
- 90° lies on the positive y-axis, which is the boundary between Quadrants I and II.
- But since it's exactly 90°, its reference angle is 0°? Wait — actually, 90° is a special case.
But reference angle is defined as the acute angle to the x-axis. For 90°, the angle to the x-axis is 90°, but that's not acute.
Wait — reference angles are measured from the x-axis to the terminal side, so:
- At 90°, the terminal side is on the y-axis → distance to x-axis is 90°, but we usually say the reference angle is 90° only if we allow it?
Actually, standard definition: The reference angle is always between 0° and 90°, inclusive.
So for 90°, the reference angle is 90°? No — wait!
Let’s clarify:
> A reference angle is the acute angle between the terminal side and the x-axis.
At 90°, the terminal side is perpendicular to the x-axis → the angle between them is 90°, but that’s not acute.
However, by convention, when the angle is 90°, 180°, 270°, or 360°, the reference angle is taken as the acute angle formed with the axis, so:
- For 90°, reference angle = 90°? No — actually, many sources define reference angle for quadrantal angles as 0° or undefined, but typically:
> Reference angle for 90° is 90°, but since it's not acute, some say it's undefined.
But more commonly, we consider reference angle for angles on axes as the smallest angle to the axis, so:
- 90° → reference angle = 90° (but not acute), but most textbooks say:
> Reference angle is the acute angle between the terminal side and the x-axis.
So at 90°, the angle between terminal side and x-axis is 90°, which is not acute, so perhaps we say the reference angle is 90°?
Wait — let’s check standard rule:
> The reference angle is the acute angle between the terminal side and the x-axis.
So if the angle is 90°, the angle between terminal side and x-axis is 90°, which is not acute, so does it have a reference angle?
No — but in practice, for quadrantal angles like 90°, 180°, etc., we often say the reference angle is 0° or just leave it as 90°, but most sources say:
> The reference angle for 90° is 90°, even though it's not acute.
But strictly speaking, reference angles are always between 0° and 90°, inclusive.
So 90° is acceptable.
But let’s go with standard textbook logic:
> For 90°, the reference angle is 90°.
But that contradicts “acute”.
Actually, many sources define reference angle as the smallest positive angle between the terminal side and the x-axis, regardless of being acute.
So:
- 90° → reference angle = 90°
But others say it's 0°?
No — better to follow this:
> Reference angle = |θ - 180°n| where n is chosen so that result is in [0°, 90°]
For 90°:
- Closest x-axis: it's on the y-axis → angle to x-axis is 90° → so reference angle = 90°
But 90° is not acute, so some sources say no reference angle.
But in practice, for quadrantal angles, we do assign reference angles:
- 90° → ref = 90°
- 180° → ref = 0°
- 270° → ref = 90°
- 360° → ref = 0°
Wait — actually, standard rule:
> For angles on axes, the reference angle is the angle between the terminal side and the nearest x-axis, so:
- 90° → to x-axis: 90° → ref = 90°
- 180° → to x-axis: 0° → ref = 0°
- 270° → to x-axis: 90° → ref = 90°
- 360° → to x-axis: 0° → ref = 0°
But again, 90° is not acute, so maybe not considered a "reference angle".
But commonly accepted:
> Reference angle for 90° is 90°
But let’s double-check with examples.
Actually, most math sources say:
> The reference angle is the acute angle formed by the terminal side and the x-axis.
So if the terminal side is on the axis, then the angle is 90° or 0°, but 90° is not acute → so no acute angle, so reference angle is not defined?
But that can’t be — we need a consistent method.
Let’s use this method:
> Reduce angle to 0° to 360°, then:
- QI: θ
- QII: 180° − θ
- QIII: θ − 180°
- QIV: 360° − θ
And for quadrantal angles, the reference angle is the smallest angle to the x-axis, which is:
- 90° → 90°
- 180° → 0°
- 270° → 90°
- 360° → 0°
But since 90° is not acute, some say reference angle is undefined, but in practice, we accept 90°.
But let’s look at 450°:
- 450° − 360° = 90°
- So 450° is coterminal with 90°
- 90° is on the positive y-axis → reference angle = 90°
But since 90° is not acute, and reference angles are usually acute, we may say reference angle is 90°, but it’s a special case.
So for 450°:
- Coterminal angle: 90°
- Reference angle: 90°
✔ Answer: 90°
---
4. 35°
- 35° is in Quadrant I
- Reference angle = 35°
✔ Answer: 35°
---
5. $ \frac{5\pi}{6} $
Convert to degrees:
$ \frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ $
- 150° is in Quadrant II
- Reference angle = 180° − 150° = 30°
✔ Answer: 30°
---
6. $ -\frac{\pi}{2} $
Convert to degrees:
$ -\frac{\pi}{2} \times \frac{180^\circ}{\pi} = -90^\circ $
Add 360°:
-90° + 360° = 270°
- 270° is on the negative y-axis
- Reference angle = angle to x-axis = 90°
(From 270° to x-axis is 90°)
✔ Answer: 90°
---
7. $ \frac{13\pi}{6} $
Convert to degrees:
$ \frac{13\pi}{6} \times \frac{180^\circ}{\pi} = \frac{13 \times 180}{6} = 390^\circ $
Reduce: 390° − 360° = 30°
- 30° is in Quadrant I
- Reference angle = 30°
✔ Answer: 30°
---
8. $ \frac{7\pi}{4} $
Convert to degrees:
$ \frac{7\pi}{4} \times \frac{180^\circ}{\pi} = \frac{7 \times 180}{4} = 315^\circ $
- 315° is in Quadrant IV
- Reference angle = 360° − 315° = 45°
✔ Answer: 45°
---
## ✔ Part 2: Find the reference angle for each angle
13. 30° α = ________
- 30° is in Quadrant I
- Reference angle = 30°
✔ Answer: 30°
---
14. -150° α = ________
- Add 360°: -150° + 360° = 210°
- 210° is in Quadrant III
- Reference angle = 210° − 180° = 30°
✔ Answer: 30°
---
15. $ \frac{7\pi}{6} $ α = ________
Convert to degrees:
$ \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = \frac{7 \times 180}{6} = 210^\circ $
- 210° is in Quadrant III
- Reference angle = 210° − 180° = 30°
✔ Answer: 30°
---
16. $ -\frac{5\pi}{2} $ α = ________
Convert to degrees:
$ -\frac{5\pi}{2} \times \frac{180^\circ}{\pi} = -\frac{5 \times 180}{2} = -450^\circ $
Now add 360° repeatedly:
- -450° + 360° = -90°
- -90° + 360° = 270°
- 270° is on the negative y-axis
- Reference angle = angle to x-axis = 90°
✔ Answer: 90°
---
## ✔ Part 3: Convert from radians to degrees or degrees to radians
17. 45° = ________ radians
Use: $ \text{radians} = \text{degrees} \times \frac{\pi}{180} $
$ 45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} $
✔ Answer: $ \frac{\pi}{4} $
---
18. -150° = ________ radians
$ -150^\circ = -150 \times \frac{\pi}{180} = -\frac{150\pi}{180} = -\frac{5\pi}{6} $
✔ Answer: $ -\frac{5\pi}{6} $
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## ✔ Final Answers Summary
Part 1: Reference Angles
1. 330° → 30°
2. -75° → 75°
3. 450° → 90°
4. 35° → 35°
5. $ \frac{5\pi}{6} $ → 30°
6. $ -\frac{\pi}{2} $ → 90°
7. $ \frac{13\pi}{6} $ → 30°
8. $ \frac{7\pi}{4} $ → 45°
Part 2: Reference Angles
13. 30° → 30°
14. -150° → 30°
15. $ \frac{7\pi}{6} $ → 30°
16. $ -\frac{5\pi}{2} $ → 90°
Part 3: Conversions
17. 45° = $ \frac{\pi}{4} $
18. -150° = $ -\frac{5\pi}{6} $
---
Let me know if you'd like this formatted as a printable answer sheet!
Parent Tip: Review the logic above to help your child master the concept of reference angles worksheet.
