How to Calculate the Cosine of an Angle
How to Calculate the Sine of an Angle
How to Measure Angles with Radians

How to Calculate Values for the Six Trigonometric Functions

In pre-calculus, you need to evaluate the six trig functions — sine, cosine, tangent, cosecant, secant, and cotangent — for a single angle on the unit circle. For each angle on the unit circle, three other angles have similar trig function values. The only difference is that the signs of these values are opposite, depending on which quadrant the angle is in. Sometimes the angle won't be on the unit circle and you'll have to use your calculator.

If you don't have the unit circle at your disposal (if you're taking a test, for instance), you can draw a picture and find the values you need the long way.

The point-in-the-plane definition of cosine in a right triangle is

image0.png

Because the hypotenuse r is always 1 in the unit circle, the x value is the cosine value. And if you remember the alternate definition of sine,

image1.png

you'll realize that the y value is the sine value. Therefore, any point anywhere on the unit circle is always

image2.png

Talk about putting all the pieces together!

Alphabetically, x comes before y and c comes before s (cosine comes before sine, in other words). This fact should help you remember which one is which.

Tangent, cotangent, secant, and cosecant require a little more effort than the sine and cosine do. For many angles on the unit circle, evaluating these functions requires some careful work with fractions and square roots. Remember to always rationalize the denominator for any fraction in your final answer. Also, remember that any number divided by 0 is undefined. The tangent and secant functions, for instance, are undefined when the cosine value is 0. Similarly, the cotangent and cosecant values are undefined when the sine value is 0.

Time for an example. To evaluate the six trigonometric functions of 225 degrees using the unit circle, follow these steps:

  1. Draw the picture.

    When you're asked to find the trig function of an angle, you don't have to draw out a unit circle every time. Instead, use your smarts to figure out the picture. For this example, 225 degrees is 45 degrees more than 180 degrees. Draw out a 45-45-90-degree triangle in the third quadrant only.

  2. Fill in the lengths of the legs and the hypotenuse.

    A 45er triangle, decked out like a Christmas tree.
    A 45er triangle, decked out like a Christmas tree.

    Use the rules of the 45er triangle. The coordinate of the point at 225 degrees is

    image4.png

    The figure shows the triangle, as well as all the information to evaluate the six trig functions.

    Be careful! Use what you know about the positive and negative axes on the coordinate plane to help you. Because the triangle is in the third quadrant, both the x and y values should be negative.

  3. Find the sine of the angle.

    The sine of an angle is the y value, or the vertical line that extends from the point on the unit circle to the x-axis. For 225 degrees, the y value is

    image5.png
  4. Find the cosine of the angle.

    The cosine value is the x value, so it must be

    image6.png
  5. Find the tangent of the angle.

    To find the tangent of an angle on the unit circle, you use the tangent's alternate definition:

    image7.png

    Another way of looking at it is that

    image8.png

    because in the unit circle, the y value is the sine and the x value is the cosine. So if you know the sine and cosine of any angle, you also know the tangent. (Thanks, unit circle!) The sine and the cosine of 225 degrees are both

    image9.png

    Therefore, you can divide the sine by the cosine to get the tangent of 225 degrees, which is 1.

  6. Find the cosecant of the angle.

    The cosecant of any angle is

    image10.png

    or r/y, using the point-in-the-plane definition. Using what you determined in Step 1,

    image11.png

    You can now divide 1 by

    image12.png
  7. Find the secant of the angle.

    The secant of any angle is

    image13.png

    Because the cosine of 225 degrees is also

    image14.png

    found in Step 4, the secant of 225 degrees is

    image15.png
  8. Find the cotangent of the angle.

    The cotangent of an angle is

    image16.png

    From Step 5, tan (225 degrees) = 1. So cot (225 degrees) = 1/1 = 1. Easy as pie!

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