How to Calculate Trigonometry Functions of Angles Using the Unit Circle - dummies

How to Calculate Trigonometry Functions of Angles Using the Unit Circle

By Mary Jane Sterling

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

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Using the angles shown, find the tangent of theta.

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

    The coordinates are

    image1.jpg
    image2.jpg

    The radius is r = 1.

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

    The ratio for the tangent is y/x, so you find that

    image3.jpg

Next, using the angles shown, find the cosine of sigma.

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

    The coordinates are

    image4.jpg
    image5.jpg

    the radius is r = 1.

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

    The ratio for the cosine is x/r, which means that you need only the x-coordinate, so

    image6.jpg

Now, using the angles shown, find the cosecant of beta.

  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.

    The ratio for cosecant is r/y, which means that you need only the y-coordinate, so

    image7.jpg