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Reference Angle Calculator


Table of Content


The reference angle calculator helps you find the acute (smaller than 90 degrees) angle in the standard position between the terminal side and the x-axis of an angle. This reference angle simplifies trigonometric calculations and helps to determine the positive or negative nature of trigonometric functions.

What is a Reference Angle in Trigonometry?

A reference angle is the positive acute angle formed between the terminal side of an angle and the nearest x-axis in a Cartesian coordinate system.

In other words, the reference angle is that angle sandwiched between the terminal side and the x-axis and it always falls between 0 and 90 degrees.

reference angle

How Do We Find The Reference Angle?

  • When the terminal side is at 0° to 90°, the reference angle is the same as the given angle.
  • When the terminal side is at angles from 90° to 180°, the reference angle is 180° minus the given angle.
  • When the terminal side is at 180° to 270°, the reference angle is the given angle minus 180°.
  • When the terminal side is at 270° to 360°, the reference angle is 360° minus the given angle.

Practical Example:

Let’s say you are hiking on a hill. The slope of the hill is 65 degrees from the horizontal ground. So, find the in terms of degree.


To find the reference angle in degrees, you can use the formula:

Reference Angle = 90 degrees - Angle of Slope

In this case, the angle of the slope is 65 degrees. Plug this value into the formula:

Reference Angle = 90 degrees - 65 degrees = 25 degrees

So, the reference angle in degrees for this hill is 25 degrees. This tells you that the slope of the hill is 25 degrees away from being completely vertical (90 degrees from the horizontal ground).

How Does This Calculator Work?

The reference angle calculator enables you to input any angle and determine its corresponding reference angle, which is an acute angle. If you don’t want to indulge in formulas and complex calculations then just try the calculator: 

What To Do:

  • Simply put the angle to find its reference angle which is the acute angle that corresponds to the angle entered. 
  • Select either degrees or radians
  • Tap "Calculate"

What You Get:

  • Angle in degrees
  • Reference angle in radians
  • Angle in pi radians
  • A graph of the angle in the coordinate system is displayed.

Additional Queries:

Why Do We Need To Find The Reference Angle?

Trigonometric functions can be evaluated for angles outside the first quadrant using reference angles. Additionally, they can be used to find the (x,y) coordinates of those angles.

Can a Reference Angle Be Negative?

In order to find the reference angle we always find the difference between 225˚ and 180˚. It is important to note that the reference angle can never be negative.

References: Reference Angle, Reference Angle Definition, How to Draw Reference Angle? Rules for Reference Angles in Each Quadrant, How to Find Reference Angles? and Steps to Find Reference Angles.

Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.

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