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The Cosine Function: Adjacent over Hypotenuse

When you’re using right triangles to define trigonometry functions, the trig function cosine, abbreviated cos, has input values that are angle measures and output values that you obtain from the ratio

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Take a look at the following figure,

A right triangle with sides of lengths 3, 4, and 5.
A right triangle with sides of lengths 3, 4, and 5.

and you see that the cosines of the two angles are

image2.png

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 dealing with circles), never greater than 1, because the hypotenuse is the denominator.

Now for an example.

image3.png

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

    image5.png

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 (complementary angles add up to 90 degrees).

image6.png

Here’s an example using the property of complementary angles:

image7.png

  1. First find the cosine of angle B.

    image8.png

    you that the side opposite angle B measures 5 units. You just have to find the measure of the side adjacent to angle B.

  2. Find the measure of the third side of the triangle.

    image9.png
  3. Use the ratio to find the sine of B.

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