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Reference Angle Calculator

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The reference angle calculator finds the angle in the first quadrant which corresponds to the given angle.

This calculator is convenient to use to find the first quadrant (or acute) angle for a given angle. It’s as easy as entering the value of the angle in degrees and the calculator will determine the reference angle.

What Exactly is a Reference Angle?

If you notice, every angle from 0° to 360° is measured from the positive x-axis to the line that ends the angle, also known as the terminal side of the angle.  The direction to the terminal side is in a counter clockwise direction. The reference angle is then the smallest angle that can be made with the x-axis and the terminal side, moving in either the counter clockwise or clockwise direction.

How Do You Find a Reference Angle?

Let’s examine how to find the reference angle.

Suppose the angle in question has a measure of 545°.

The rule is that if the angle is greater than 360°, subtract 360 in succession until the angle is smaller than 360°. In our scenario 545 – 360 = 185.

Next, locate the angle in either quadrant I, II, III or IV.

Use the following rules to determine the reference angle, given the quadrant.

1st quadrant: reference angle = angle

2nd quadrant: reference angle = 180° – angle

3rd quadrant: reference angle = angle – 180°

4th quadrant: reference angle = 360 – angle

Therefore, in this example the reference angle is 185 – 180 = 5°.

When dealing with trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant, most of the time they give the identical value for the angle and its reference angle. Make sure you are aware of the signs, which change depending on the quadrant for which the angle falls.

Sine is positive in the 1st and 2nd quadrant. Cosine is positive in the 1st and 4th quadrant.  The signs of the other trigonometric functions can be obtained from the knowledge of sine and cosine. Review the trigonometric ratios for the other trig functions if needed.

If you need to manually find the reference angle, the calculator will be handy to check your work. Or, simply skip the manual work and get quick and accurate results with this calculator.