Reference Angle Calculator

Angle
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Reference angle
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Our reference angle calculator is a handy tool for recalculating angles into their acute version. All you have to do is simply input any angle from 0° to 360°, and this calculator will find the reference angle for you.

This article explains what exactly is a reference angle. It will also provide you with a step-by-step instruction how to find a reference angle, along with a few examples.

What is a reference angle

Look at the picture above. Every angle is measured from the positive part of x-axis to its terminal line (the line that determines the end of the angle) counterclockwise. If you want to find the reference angle, you have to find the smallest possible angle formed by the x-axis and the terminal line, going either clockwise or counterclockwise.

Reference angles are useful in trigonometry. If you want to find the sine or cosine of any arbitrary angle, you have to first find its reference angle in the first quarter. Then, you can find the trigonometric function of the reference angle and choose a proper sign.

How to find the reference angle

It's easier than it looks! All you have to do is follow these steps:

1. Choose your initial angle - for example, 610°.
2. If your angle is larger than 360° (a full angle), subtract 360°. Keep doing it until you get an angle smaller than a full angle. In this example, after subtracting 360° we get 250°.
• 0° to 90° - first quadrant,
• 90° to 180° - second quadrant,
• 180° to 270° - third quadrant,
• 270° to 360° - fourth quadrant.

In this case, 250° lies in the third quadrant.

1. Choose a proper formula for calculating the reference angle:
• 0° to 90°: `reference angle = angle`,
• 90° to 180°: `reference angle = 180° - angle`,
• 180° to 270°: `reference angle = angle - 180°`,
• 270° to 360°: `reference angle = 360° - angle`.

In this case, we need to choose the formula `reference angle = angle - 180°`.

1. Substitute your angle to find the reference angle:

`reference angle = 250° - 180° = 70°`.

Trigonometric functions

Generally, trigonometric functions (sine, cosine, tangent, cotangent) give the same value for both an angle and its reference angle. The only thing that changes is the sign - these functions are positive and negative in various quadrants. Follow the "All Students Take Calculus" mnemonic rule (ASTC) to remember when these functions are positive.

• A for all: in the first quadrant, all trigonometric functions have positive values.
• S for sine: in the second quadrant, only the sine function has positive values.
• T for tangent: in the third quadrant, tangent and cotangent have positive values.
• C for cosine: in the fourth quadrant, the cosine function has positive values.

Make sure to take a look at our law of cosines calculator for more information about trigonometry.