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How to Calculate Trigonometry Functions of Angles Using the Unit Circle

Calculating trig functions on angles using the unit circle is easy as pie. The following figure shows a unit circle, which has the equation x2 + y2 = 1, along with some points on the circle and their coordinates.

image0.jpg

Using the angles in the figure, find the tangent of the angle θ.

  1. Find the x- and y-coordinates of the point where the angle’s terminal side intersects with the circle.

    image1.png
  2. Determine the ratio for the function and substitute in the values.

    image2.png

Next, using the angles in the preceding figure, find the cosine of the angle σ.

  1. Find the x- and y-coordinates of the point where the terminal side of the angle intersects with the circle.

    image3.png
  2. Determine the ratio for the function and substitute in the values.

    image4.png

Now, using the angles in the figure, find the cosecant of the angle β.

  1. Find the x- and y-coordinates of the point where the terminal side of the angle intersects with the circle.

    The coordinates are x = 0 and y = –1; the radius is r = 1.

  2. Determine the ratio for the function and substitute in the values.

    image5.png
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