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The Cosine Function: Adjacent over Hypotenuse

The trig function cosine, abbreviated cos, works by forming this ratio: adjacent/hypotenuse. In the figure, you see that the cosines of the two angles are as follows:

image0.jpg image1.jpg

The situation with the ratios is the same as with the sine function — the values are going to be less than or equal to 1 (the latter only when your triangle is a single segment or when dealing with circles), never greater than 1, because the hypotenuse is the denominator.

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The two ratios for the cosine are the same as those for the sine — except the angles are reversed. This property is true of the sines and cosines of complementary angles in a right triangle (meaning those angles that add up to 90 degrees).

If theta and lambda are the two acute angles of a right triangle, then sin theta = cos lambda and cos theta = sin lambda.

Now for an example. To find the cosine of angle beta in a right triangle if the two legs are each

image3.jpg

feet in length:

  1. Find the length of the hypotenuse.

    Using the Pythagorean theorem, a2 + b2 = c2, and replacing both a and b with the given measure, solve for c.

    image4.jpg

    The hypotenuse is

    image5.jpg

    feet long.

  2. Use the ratio for cosine, adjacent over hypotenuse, to find the answer.

    image6.jpg
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