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Trigonometric Ratios: Cosine - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Right triangles have ratios that are used to represent their base angles. Cosine ratios, along with sine and tangent ratios, are ratios of two different sides of a right triangle. Cosine ratios are specifically the ratio of the side adjacent to the represented base angle over the hypotenuse. In order to find the measure of the angle, we must understand inverse trigonometric functions.

In right triangles there exist special relationships between interior angles and the sides. We're going to talk about cosine and the special relationship that cosine gives you. If we talked about an angle theta and again we could chose one of these two angles and then that would determine how we would describe theta. So theta is just an angle and cosine is a ratio between the side that is adjacent to it which means next to it but not the hypotenuse and the hypotenuse. So cosine is the ratio of the adjacent side to the hypotenuse and one way to remember this is using the SOH CAH TOA saying. I don't know how many years I've been saying this in Geometry classes but SOH CAH TOA is an easy way to remember your 3 trigonometric ratios. Sine, cosine tangent where sine is the ratio of the opposite side to the hypotenuse, cosine is ratio of adjacent side to hypotenuse and tangent is ratio of opposite side to adjacent.
Let's look at one quick example to see how we can use cosine to come up with ratios. We talked about cosine of s, s is the ratio of the adjacent side which is r to the hypotenuse which is t. So again cosine is a ratio of two different side links. Cosine of r so now we're going to forget about this vertex s and we're going to look at vertex r. From vertex r's perspective the adjacent side is s and the hypotenuse is still going to be t. So you can use cosine to describe the relationship between an angle, its adjacent side and the hypotenuse.

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