In a right triangle, there are 3 trigonometric ratios: sine, cosine and tangent. Each ratio has its own reciprocal and function. It is with these ratios that the unit circle is constructed.
Trigonometric ratios in right angle triangles express a relationship between the length of two sides.
Consider the right triangle ABC below:
The different trigonometric ratios are:||\begin{align}\sin\,(\angle A)&=\dfrac{\text{Leg opposite to}\ \angle A}{\text{Hypotenuse}}\\[2pt]&=\dfrac{a}{c}\\[10pt]\cos\,(\angle A)&=\dfrac{\text{Leg adjacent to}\ \angle A}{\text{Hypotenuse}}\\[2pt]&=\dfrac{b}{c}\\[10pt]\tan\,(\angle A)&=\dfrac{\text{Leg opposite to}\ \angle A}{\text{Leg adjacent to}\ \angle A}\\[2pt]&=\dfrac{a}{b}\end{align}||
Some trigonometric ratios are equivalent, such that by choosing the appropriate angle and ratio, we get the same value.
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\,(\angle A)=\dfrac{a}{c}=\cos\,(\angle 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
To validate your understanding of trigonometry, see the following interactive Crash Course:
