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Trigonometric Ratios in Right Triangles
Concept sheet | Mathematics
100%
100%
Secondary
4-5
Table of contents
Trigonometric ratios in right angle triangles express a relationship between the lengths of two sides.
The trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. The first three are discussed in the following concept sheet, while the other three will be discussed in the concept sheets on the unit circle.
Consider the right triangle ABC below:
The different trigonometric ratios are:
||\begin{align} \sin A &= \dfrac{\text{Measure of the leg opposite angle A}}{\text{Measure of the hypotenuse}}\\ &= \dfrac{a}{c}\\\\ \cos A &= \dfrac{\text{Measure of the adjacent leg to angle A}}{\text{Measure of the hypotenuse}} \\ &= \dfrac{b}{c}\\\\ \tan A &= \dfrac{\text{Measure of the leg opposite angle A}}{\text{Measure of the leg adjacent to angle A}} \\ &= \dfrac{a}{b}\\\\ \text{cosec } A &= \dfrac{\text{Measure of the hypotenuse}}{\text{Measure of the leg opposite angle A}} \\ &= \dfrac{c}{a}\\\\ \sec A &= \dfrac{\text{Measure of the hypotenuse}}{\text{Measure of leg adjacent to angle A}} \\ &= \dfrac{c}{b}\\\\ \text{cotan } A &= \dfrac{\text{Measure of leg adjacent to angle A}}{\text{Measure of leg opposite angle A}} \\ &= \dfrac{b}{a}\end{align}||
In addition, some trigonometric ratios are equivalent. Therefore, by choosing the angle and the appropriate ratio, we can obtain equivalent values.
In a right angle triangle, the sine of one acute angle is equal to the cosine of the other acute angle. For example, observe the ratios in the triangle below: ||\sin A = \dfrac{a}{c} = \cos B||
There is a mnemonic trick to help identify the basic trigonometric ratios with sine, cosine, and tangent.
Just remember the expression SOH - CAH - TOA.
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent