Finding a Reference Angle
Reference Angle Calculator: Do you really want to calculate the reference angle for any known angle? If yes, then take the advantge of the user-friendly Finding REference Angle Tool that produces the accurate results in a short span of time. Continue reading to learn the more details about the topic such are what is a reference angle, how to find reference angle in degrees or radians. Get the example questions with solutions and rules for reference angle in each quadrant.
What is a Reference Angle?
The reference angle is the smallest possible angle made by the terminal side of the given angle with the x-axis. It is always an acute angle and it is always positive irrespective of the side of the axis it is falling.
The commonly used angles and trigonometric functions are given in the tabular format.
| α(°) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| α(rad) | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| sin(α) | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 | −1 | 0 |
| cos(α) | 1 | √3/2 | √2/2 | 1/2 | 0 | −1 | 0 | 1 |
| tan(α) | 0 | √3/3 | 1 | √3 | - | 0 | - | 0 |
| cot(α) | - | √3 | 1 | √3/3 | 0 | - | 0 | 0 |
Reference Angle Formula
There are 4 quandrants in coordinate system. They are I, II, III and IV quadrants. Trigonometric functions give the same value for both angles and regerence angles. It only changes sign. Follow ASTC rule to remember when these functions are positive.
- A is all: in the first quadrant, all trig functions are posistive.
- S is Sine: in the second quadrant, only the sine function has positive values.
- T for tangent: in the third quadrant, tan and cot have positive values.
- C for cosine: in the fourth quadrant,only sos function has posistive values.
The reference angle formulas depending on the quadrant of the given angle are here. WE use same rule even for radians.
| Quadrant | Angle, θ | Reference Angle Formula in Degrees | Reference Angle Formula in Radians |
|---|---|---|---|
| I | lies between 0° and 90° | θ | θ |
| II | lies between 90° and 180° | 180 - θ | π - θ |
| III | lies between 180° and 270° | θ - 180 | θ - π |
| IV | lies between 270° and 360° | 360 - θ | 2π - θ |
Steps to Find Reference Angle
The easy steps on how to calculate the reference angle are listed here. Go through these guidelines to get the instant results.
- Determine the coterminal angle for the given angle that lies between 0° and 360°.
- If the angle lies between 0° and 90°, then angle is the given angle reference angle. Otherwise, check whether it is closest to 180° or 360° and by how much.
- The angle from above step is the reference angle for the angle.
Example:
Find the reference angle of 480°.
Solution:
Given angle is 480°
The coterminal angle is 480° - 360° = 120°
Angle is closest to 180° or 360°
180° - 120° = 60°
The reference angle of 480° is 60°.
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FAQ's on Finding Reference Angle Calculator With Steps
1. How to find reference angle?
The steps to calculate the reference angle are here:
- Firstly, find the coterminal angle for the given angle that lies between 0° to 360°.
- Check whether the obtained angle is close to 180° or 360° and by how much.
- Now, obtained is the reference angle of the given angle.
2. Can reference angles be negative?
A reference angle is a non-negative angle. It is always positive and can never be negative in measurement.
3. What is the reference angle for 7π/6?
The reference angle for 7π/6 is π/6.
4. How to draw reference angles?
To draw a reference angle for an angle, identify its terminal side and see by what angle the terminal side is close to the x-axis.
