When you want to recalculate angles into their acute version, then a reference angle calculator can be the best tool to have. Once you have a positive angle, you can input the data and find out the reference angle almost immediately. Below, you can learn all about reference angles, what they are, and how you can find them.

What is a Reference Angle?

A reference angle is a small angle, the smallest in fact, that forms at the X-axis and terminal line in a clockwise or anti-clockwise direction. What you will possibly know is that you measure every angle from the positive area of the X-axis to the terminal line. The terminal line is what forms the end of an angle. 

Most people use reference angles in trigonometry, especially if they are looking for the sine or cosine of an arbitrary angle. The first step in that process would be to identify the reference angle before getting to the trigonometric function of the reference angle.

In the table below, you will see the most-common angles and their trigonometric functions.



a(°)
0°
30°
45°
60°
90°
180°
270°
360°


a(rad)
0
π/6
π/4
π/3
π/2
Π
3 π/2
2 π


sin(α)
0
½
√2/2
√2/3
1
0
-1
0


cos(α)
1
√2/3
√2/2
½
0
-1
0
1


tg(α)
0
√3/3
1
√3
-
0
-
0


ctg(α)
-
√3
1
√3/3
0
-
0
-

Trigonometric Functions and Graph Quadrants

A 2D Cartesian system’s two axes divide a plane into quadrants, or regions. Numbering starts in the top right-hand corner with positive coordinates and an anti-clockwise direction.



You get the same value for the angle and reference angle from all trigonometric functions – cotangent, tangent, sine, and cosine. The only thing that differs is the sign, whether the quadrants have negative or positive coordinates. 

The way to remember is by using the ASTC rule – All Students Take Calculus. 

A: All – All trigonometric functions have positive values in the first quadrant 
S: Sine – Sine is in the second quadrant with positive values
T: Tangent – Tangent and cotangent have positive values in the third quadrant
C: Cosine – in the fourth quadrant, the cosine function has positive values

How to Find a Reference Angle in Degrees

Finding a reference angle in degrees is straightforward if you follow the correct steps. 

   1.	Identify your initial angle. For this example, we’ll use 440°

   2.	The angle is larger than a full angle of 360°, so you need to subtract the total angle until it’s small. 
440° - 360° = 80°

   3.	Find out the quadrant your angle is in  
0° to 90° - first
90° to 180° - second
180° to 270° - third
270° to 360° - fourth

In this example, the angle is in the first quadrant.

   4.	Choose the reference angle formula to suit your quadrant and angle: 
0° to 90°: reference angle = the angle
90° to 180°: reference angle = 180° - the angle
180° to 270°: reference angle = the angle - 180°
270° to 360°: reference angle = 360° - the angle

In this instant, the reference angle = the angle

   5.	Input your angle data to find the reference angle 
reference angle = 80°

How to Find a Reference Angle in Radians

Finding your reference angle in radians is similar to identifying it in degrees. 

1.	Find your angle. For this example, we’ll use 28π/9

2.	If your angle is larger than 2π, take away the multiples of 2π until you get a value that’s smaller than the full angle. 
10π9

3.	Identify the quadrants: 
0 to π/2 - first quadrant, meaning reference angle = angle
π/2 to π - second quadrant, meaning reference angle = π – angle
π to 3π/2 - third quadrant, meaning reference angle = angle – π
3π/2 to 2π - fourth quadrant, meaning reference angle = 2π – angle

Our example of 10π/9 is more than π, so we will put it into the third quadrant.
Angle = the angle – π = π/9

How to Use a Reference Angle Calculator?

If you would rather avoid all the formulas associated with identifying a reference angle and, instead, get your answer quickly, then you will need a reference angle calculator. Using one is effortless. 

   1.	Identify your positive angle and type it into the box. You can change the units to degrees, radians, or any number of angle types. 

   2.	Get your answer! The reference angle calculator does all the hard work for you.

Commonly Asked Questions: What is the Reference Angle for…

Often, you will want a quick answer to a burning question about reference angles. Below, you can find the reference angles in radian and degrees. 

Calculated Reference Angle - 1°: 1°
Calculated Reference Angle - 2°: 2°
Calculated Reference Angle - 3°: 3°
Calculated Reference Angle - 4°: 4°
Calculated Reference Angle - 5°: 5°
Calculated Reference Angle - 6°: 6°
Calculated Reference Angle - 7°: 7°
Calculated Reference Angle - 8°: 8°
Calculated Reference Angle - 9°: 9°
Calculated Reference Angle - 10°: 10°
Calculated Reference Angle - 15°: 15°
Calculated Reference Angle - 20°: 20°
Calculated Reference Angle - 25°: 25°
Calculated Reference Angle - 30°: 30° (π / 6)
Calculated Reference Angle - 35°: 35°
Calculated Reference Angle - 40°: 40°
Calculated Reference Angle - 45°: 45° (π / 4)
Calculated Reference Angle - 50°: 50°
Calculated Reference Angle - 55°: 55°
Calculated Reference Angle - 60°: 60° (π / 3)
Calculated Reference Angle - 65°: 65°
Calculated Reference Angle - 70°: 70°
Calculated Reference Angle - 75°: 75°
Calculated Reference Angle - 80°: 80°
Calculated Reference Angle - 85°: 85°
Calculated Reference Angle - 90°: 90° (π / 2)
Calculated Reference Angle - 95°: 85°
Calculated Reference Angle - 100°: 80°
Calculated Reference Angle - 105°: 75°
Calculated Reference Angle - 110°: 70°
Calculated Reference Angle - 115°: 65°
Calculated Reference Angle - 120°: 60° (π / 3)
Calculated Reference Angle - 125°: 55°
Calculated Reference Angle - 130°: 50°
Calculated Reference Angle - 135°: 45° (π / 4)
Calculated Reference Angle - 140°: 40°
Calculated Reference Angle - 145°: 35°
Calculated Reference Angle - 150°: 30° (π / 6)
Calculated Reference Angle - 155°: 25°
Calculated Reference Angle - 160°: 20°
Calculated Reference Angle - 165°: 15°
Calculated Reference Angle - 170°: 10°
Calculated Reference Angle - 175°: 5°
Calculated Reference Angle - 180°: 0°
Calculated Reference Angle - 185°: 5°
Calculated Reference Angle - 190°: 10°
Calculated Reference Angle - 195°: 15°
Calculated Reference Angle - 200°: 20°
Calculated Reference Angle - 205°: 25°
Calculated Reference Angle - 210°: 30° (π / 6)
Calculated Reference Angle - 215°: 35°
Calculated Reference Angle - 220°: 40°
Calculated Reference Angle - 225°: 45° (π / 4)
Calculated Reference Angle - 230°: 50°
Calculated Reference Angle - 235°: 55°
Calculated Reference Angle - 240°: 60° (π / 3)
Calculated Reference Angle - 245°: 65°
Calculated Reference Angle - 250°: 70°
Calculated Reference Angle - 255°: 75°
Calculated Reference Angle - 260°: 80°
Calculated Reference Angle - 265°: 85°
Calculated Reference Angle - 270°: 90° (π / 2)
Calculated Reference Angle - 275°: 85°
Calculated Reference Angle - 280°: 80°
Calculated Reference Angle - 285°: 75°
Calculated Reference Angle - 290°: 70°
Calculated Reference Angle - 295°: 65°
Calculated Reference Angle - 300°: 60° (π / 3)
Calculated Reference Angle - 305°: 55°
Calculated Reference Angle - 310°: 50°
Calculated Reference Angle - 315°: 45° (π / 4)
Calculated Reference Angle - 320°: 40°
Calculated Reference Angle - 325°: 35°
Calculated Reference Angle - 330°: 30° (π / 6)
Calculated Reference Angle - 335°: 25°
Calculated Reference Angle - 340°: 20°
Calculated Reference Angle - 345°: 15°
Calculated Reference Angle - 350°: 10°
Calculated Reference Angle - 355°: 5°
Calculated Reference Angle - 360°: 0°