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How to Find the Trigonometry Function of an Angle in a Unit Circle

You can determine the trig functions for any angles that relate to the unit circle. To do this, you can use the rules for reference angles, the values of the functions of certain acute angles, and the rule for the signs of the functions. These functions also need to be graphed in standard position (meaning the vertex of the angle is at the origin, and the initial side lies along the positive x-axis).

Use these tables and graph to find function values by using reference angles.
Use these tables and graph to find function values by using reference angles.

The preceding figure has all the information you need.

Now, armed with all the necessary information, find the tangent of 300 degrees:

  1. Find the reference angle.

    Using the figure, you can see that a 300-degree angle is in the fourth quadrant, so you find the reference angle by subtracting 300 from 360. Therefore, the measure of the reference angle is 60 degrees.

  2. Find the numerical value of the tangent.

    Using the figure, you see that the numerical part of the tangent of 60 degrees is

    image1.png
  3. Find the sign of the tangent.

    Because a 300-degree angle is in the fourth quadrant, and angles in that quadrant have negative tangents, the tangent of 300 degrees is

    image2.png

To try your hand at working with radians, find the cosecant of

image3.png
  1. Find the reference angle.

    Using the preceding figure, this angle is in the third quadrant, so you find the reference angle by subtracting

    image4.png
  2. Find the numerical value of the cosecant.

    In the figure, the cosecant doesn’t appear. However, the reciprocal of the cosecant is sine. So find the value of the sine, and use its reciprocal.

    image5.png
  3. Find the sign of the cosecant.

    In the third quadrant, the cosecant of an angle is negative, so the cosecant of

    image6.png
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