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.
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.
It's easier than it looks! All you have to do is follow these steps:
In this case, 250° lies in the third quadrant.
reference angle = angle,
reference angle = 180° - angle,
reference angle = angle - 180°,
reference angle = 360° - angle.
In this case, we need to choose the formula
reference angle = angle - 180°.
reference angle = 250° - 180° = 70°.
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.
Make sure to take a look at our law of cosines calculator for more information about trigonometry.