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The reference angle is a positive, pointed angle formed by the extreme side of the given angle and the x-axis. To find the reference angle, determine in which quadrant the given angle lies on the coordinate plane. Then apply the appropriate reference angle formula based on the quadrant the angle is in. Read below to review what reference angles are, how to find them in degrees and radians, and what to do when an angle is negative or greater than 360° (or 2𝛑)!
Find which quadrant is the given angle θ {\displaystyle {\theta }}
is inside. If in Q1, the reference angle θ 1 {\displaystyle {\theta }^{1}}
is inside. If in Q1, the reference angle θ 1 {\displaystyle {\theta }^{1}}is the same as θ {\displaystyle {\theta }}
is the same as θ {\displaystyle {\theta }}. If in Q2, subtract θ {\displaystyle {\theta }}
of 180°. If in Q3, subtract 180° from θ {\displaystyle {\theta }}
. If in Q4, subtract θ {\displaystyle {\theta }}
from 360°.
Determine in which quadrant the given angle is located. The coordinate plane, or the intersection between the x-axis and the y-axis, is divided into 4 quadrants that extend from 0° to 360° (or 0 to 2𝛑, if the angle is in radians). Look at the angle given to you and, based on its value, determine in which quadrant it lies.[3]
- 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.
Remember the unit circle to help you find the reference angle when expressed in radians.
If the given angle is in quadrant 1, the reference angle is the same. When the angle you are given, θ {\displaystyle {\theta }}
lies in the first quadrant, the reference angle, θ 1 {\displaystyle {\theta }^{1}}
is equal to the given angle.[4]
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
if your angle is θ {\displaystyle {\theta }}
= 40°.
- 40° is in the first quadrant, so the reference angle is θ 1 {\displaystyle {\theta }^{1}}
is also 40°.
- 40° is in the first quadrant, so the reference angle is θ 1 {\displaystyle {\theta }^{1}}
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
If the given angle is in quadrant 2, subtract the angle from 180°. When the angle you are given lies in the second quadrant, you subtract its value from 180° to get the reference angle, or θ 1 = 180 − θ {\displaystyle {\theta }^{1}=180-{\theta } }
. If the angle is in radians, subtract the angle from 𝛑 or θ 1 = π − θ {\displaystyle {\theta }^{1}={\pi }-{\theta }}
.[5]
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
if your angle is θ {\displaystyle {\theta }}
= 120°.
- 120° is in the second quadrant.
- 180° – 120° = 60°. The reference angle is θ 1 {\displaystyle {\theta }^{1}}
= 60°.
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
If the given angle is in quadrant 3, subtract 180° from the angle. When the angle you are given is in the third quadrant, you subtract 180° from the angle to get the reference angle, or θ 1 = θ − 180 {\displaystyle {\theta }^{1}={\theta }-180 }
. If the angle is in radians, subtract 𝛑 from the angle or θ 1 = θ − π {\displaystyle {\theta }^{1}={\theta }-{\pi }}
.[6]
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
if your angle is θ {\displaystyle {\theta }}
= 230°.
- 230° is in the third quadrant.
- 230° – 180° = 50°. The reference angle is θ 1 {\displaystyle {\theta }^{1}}
= 50°.
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
If the given angle is in quadrant 4, subtract the angle from 360°. When the angle you are given is in the fourth quadrant, subtract the angle from 360° to get the reference angle, or θ 1 = 360 − θ {\displaystyle {\theta }^{1}=360-{\theta }}
. If the angle is in radians, subtract the angle from 2𝛑 or θ 1 = 2 π − θ {\displaystyle {\theta }^{1}=2{\pi }-{\theta }}
.[7]
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
if your angle is θ {\displaystyle {\theta }}
= 325°.
- 325° is in the fourth quadrant.
- 360° – 325° = 35°. The reference angle is θ 1 {\displaystyle {\theta }^{1}}
= 35°.
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
. If the angle is in radians, subtract the angle from 𝛑 or θ 1 = π − θ {\displaystyle {\theta }^{1}={\pi }-{\theta }}
.[5]
. If the angle is in radians, subtract 𝛑 from the angle or θ 1 = θ − π {\displaystyle {\theta }^{1}={\theta }-{\pi }}
.[6]
. If the angle is in radians, subtract the angle from 2𝛑 or θ 1 = 2 π − θ {\displaystyle {\theta }^{1}=2{\pi }-{\theta }}
.[7]Add or subtract 360° until the given angle is between 0° and 360°. Sometimes you need to find a reference angle for a given angle that is less than 0 or greater than 360° (if in radians, less than 0 or greater than 2𝛑). Finding the reference angle is still possible; you just have to first find its corresponding angle which is between 0° and 360° (or between 0 and 2𝛑, if the angle is in radians).[8]
- 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’s 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’s 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.
Determine in which quadrant the new given angle is located. After adding or subtracting a multiple of 360° (or 2𝛑) from the given angle, find out where it now lies on the coordinate plane. Remember that:[9]
- 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.
Find the reference angle based on the quadrant in which the given angle is located. Apply the formula to find the reference angle based on which quadrant the given angle is in.[10]
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
if your angle is θ {\displaystyle {\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°.
- Reference angle for θ {\displaystyle {\theta }}
= -210° is θ 1 {\displaystyle {\theta }^{1}}
= 30°.
-
For examplefind the reference angle θ 1 {\displaystyle {\theta }^{1}}
Categories: How to
Source: HIS Education