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Easy Steps to Find the Reference Angle in Degrees & Radians

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Plus, what to do when angles are negative & greater than 360° or 2𝛑

Co-authored by Devin McSween Reviewed by Joseph Meyer

Last Updated: March 25, 2024 Fact Checked

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What is the reference angle?

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Calculating the Reference Angle

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Negative Angles & Angles > 360° (2𝛑)

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Why is the reference angle useful?

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The reference angle is the positive, acute angle that forms from a given angle’s terminal side and the x-axis. To find the reference angle, determine which quadrant the given angle lies in on the coordinate plane. Then, apply the appropriate reference angle formula based on the quadrant the angle is in. Read on below to review what reference angles are, how to find them in degrees and radians, and what to do when the angle is negative or greater than 360° (or 2𝛑)!

Determining the Reference Angle

Find what quadrant the given angle ${\theta }$ is in. If it’s in Q1, the reference angle ${\theta }^{1}$ is the same as ${\theta }$ . If it’s in Q2, subtract ${\theta }$ from 180°. If it’s in Q3, subtract 180° from ${\theta }$ . If it’s in Q4, subtract ${\theta }$ from 360°.

Section 1 of 4:

What is the reference angle?

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The reference angle is made by the x-axis and terminal side of an angle. Every angle has an initial side, which is the ray that falls on the x-axis, and a terminal side, which is the angle’s other ray. ^{[1]
X
Research source} The reference angle is the small angle formed by a given angle’s terminal side and the x-axis. ^{[2]
X
Research source}
- Note : Reference angles are always positive and less than or equal to 90°.
- A ray is a straight line with 1 endpoint.

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Section 2 of 4:

Calculating the Reference Angle

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1
Determine what quadrant the given angle is in. The coordinate plane, or the intersection between the x-axis and y-axis, is split into 4 quadrants that span from 0° to 360° (or 0 to 2𝛑, if the angle is in radians). Look at the angle given to you and determine which quadrant it lies in based on its value. ^{[3]
X
Research source}
- Quadrant 1 : Angles are between 0° to 90° or 0 to 𝛑/2.
- Quadrant 2 : Angles are between 90° to 180° or 𝛑/2 to 𝛑.
- Quadrant 3 : Angles are between 180° to 270° or 𝛑 to 3𝛑/2.
- Quadrant 4 : Angles are between 270° to 360° or 3𝛑/2 to 2𝛑.
- Memorize the unit circle to make finding the reference angle easier when it’s in radians.
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2
If the given angle is in quadrant 1, the reference angle is the same. When the angle given to you, ${\theta }$ , lies in the first quadrant, the reference angle, ${\theta }^{1}$ , is the same as the given angle. ^{[4]
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Research source}
- For example , find the reference angle ${\theta }^{1}$ if your given angle is ${\theta }$ = 40°.
  - 40° is in the first quadrant, so the reference angle ${\theta }^{1}$ is also 40°.
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3
If the given angle is in quadrant 2, subtract the angle from 180°. When the angle given to you rests in the second quadrant, you subtract its value from 180° to get the reference angle, or ${\theta }^{1}=180-{\theta }$ . If the angle is in radians, subtract the angle from 𝛑, or ${\theta }^{1}={\pi }-{\theta }$ . ^{[5]
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Research source}
- For example , find the reference angle ${\theta }^{1}$ if your given angle is ${\theta }$ = 120°.
  - 120° is in the second quadrant.
  - 180° - 120° = 60°. The reference angle is ${\theta }^{1}$ = 60°.
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4
If the given angle is in quadrant 3, subtract 180° from the angle. When the angle given to you is in the third quadrant, you subtract 180° from the angle to get the reference angle, or ${\theta }^{1}={\theta }-180$ . If the angle is in radians, subtract 𝛑 from the angle, or ${\theta }^{1}={\theta }-{\pi }$ . ^{[6]
X
Research source}
- For example , find the reference angle ${\theta }^{1}$ if your given angle is ${\theta }$ = 230°.
  - 230° is in the third quadrant.
  - 230° - 180° = 50°. The reference angle is ${\theta }^{1}$ = 50°.
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5
If the given angle is in quadrant 4, subtract the angle from 360°. When the angle given to you is in the fourth quadrant, subtract the angle from 360° to get the reference angle, or ${\theta }^{1}=360-{\theta }$ . If the angle is in radians, subtract the angle from 2𝛑, or ${\theta }^{1}=2{\pi }-{\theta }$ . ^{[7]
X
Research source}
- For example , find the reference angle ${\theta }^{1}$ if your given angle is ${\theta }$ = 325°.
  - 325° is in the fourth quadrant.
  - 360° - 325° = 35°. The reference angle is ${\theta }^{1}$ = 35°.

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Section 3 of 4:

Finding the Reference Angle for Negative Angles & Angles Greater than 360° (2𝛑)

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1
Add or subtract 360° until the given angle is between 0° and 360°. Sometimes, you have to find the reference angle for a given angle that’s less than 0 or greater than 360° (if it’s in radians, less than 0 or greater than 2𝛑). Finding the reference angle is still possible; you just first have to find its corresponding angle that’s between 0° and 360° (or between 0 and 2𝛑, if the angle is in radians). ^{[8]
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- If the angle is negative , keep adding 360° until it is between 0° and 360°. If the angle is in radians, keep adding 2𝛑 until it is between 0 and 2𝛑.
- If the angle is greater than 360° , keep subtracting 360° until it is between 0° and 360°. If the angle is in radians, subtract 2𝛑 until it is between 0 and 2𝛑.
- For example :
  - If the given angle is -210°, add 360°. -210° + 360° = 150°.
  - If the given angle is 545°, subtract 360°. 545° - 360° = 185°.
  - If the given angle is -11𝛑/6, add 2𝛑. -11𝛑/6 + 12𝛑/6 = 𝛑/6.
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2
Determine which quadrant the new given angle is in. After adding or subtracting multiples of 360° (or 2𝛑) from the given angle, find out where it now lies on the coordinate plane. Remember that: ^{[9]
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Research source}
- Angles between 0° to 90° or 0 to 𝛑/2 are in quadrant 1.
- Angles between 90° to 180° or 𝛑/2 to 𝛑 are in quadrant 2.
- Angles between 180° to 270° or 𝛑 to 3𝛑/2 are in quadrant 3.
- Angles between 270° to 360° or 3𝛑/2 to 2𝛑 are in quadrant 4.
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3
Find the reference angle based on the quadrant the given angle is in. Apply the formula to find the reference angle based on what quadrant the given angle is in. ^{[10]
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Research source}
- For example , find the reference angle ${\theta }^{1}$ if your given angle is ${\theta }$ = -210°.
  - Add 360°. -210° + 360° = 150°.
  - 150° is in quadrant 2.
  - To find the reference angle in quadrant 2, subtract the angle from 180°.
  - 180° - 150° = 30°.
  - The reference angle for ${\theta }$ = -210° is ${\theta }^{1}$ = 30°.

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Section 4 of 4:

Why is the reference angle useful?

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The trig function is the same for the given angle and reference angle. When finding the sine, cosine, tangent, or other trigonometric function of an angle, the value is the same for the given angle and the reference angle. This comes in handy when the given angle is an uncommon angle but the reference angle is an angle you already know the answer to, like 30° or 45°. ^{[11]
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- For instance, you might be asked to find the value of sine 210°. You likely don’t know the answer off the top of your head, but you can easily determine that the reference angle of 210° is 30°. Sine 30° is ½, so sine 210° is -½.
- Note : The sine might be different for the given and reference angle. Each trig function is positive or negative, depending on the quadrant the angle is in:
  - Quadrant 1 : All functions are positive.
  - Quadrant 2 : Only sine is positive.
  - Quadrant 3 : Only tangent is positive.
  - Quadrant 4 : Only cosine is positive.

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About This Article

Reviewed by:

Math Teacher

This article was reviewed by Joseph Meyer and by wikiHow staff writer, Devin McSween . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 5,074 times.

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Co-authors: 3

Updated: March 25, 2024

Views: 5,074

Categories: Geometry

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