Right triangles have ratios that are used to represent their base angles. Tangent ratios, along with cosine and sine ratios, are ratios of two different sides of a right triangle. Tangent ratios are the ratio of the side opposite to the side adjacent the angle they represent. In order to find the measure of the angle itself, one must understand inverse trigonometric functions.
In right triangles tangent which is abbreviated tan is a special relationship between the side that is opposite the angle and the side that is adjacent. So if we chose one of these angles in this right triangle, the side that is opposite is going to be over here so I'm going to say opposite side and the adjacent side is the one that is next to it but is not the hypotenuse. So I'm going to say adjacent so tangent of angle theta is the ratio of the opposite side to the adjacent side. So the way that you can keep sine, cosine and tangent together is the saying SOH CAH TOA which means sine is the ration of opposite to hypotenuse, cosine is the ration of adjacent to hypotenuse and tangent is the ratio of opposite of adjacent. So let's apply what we know about the tangent of an angle, in this right triangle here I'm asking to find tangent of angle s and tangent of angle r.
Tangent of angle s means that the opposite side which is sin ratio to the adjacent side which is r so tangent of angle s is the ratio of s:r. To find tangent of r and again I'm now pronouncing this tan it's still tangent is if I look at angle r the opposite side is lower case r and the adjacent is s. So notice that in a right triangle the tangents of the opposite angles are going to be reciprocals of each other and in any right triangle, tangent is the relationship of the opposite side to the adjacent side.