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Trigonometric Ratios

"Trigon" is Greek fortriangle, and "metric" is Greek for measurement. Thetrigonometric ratiosare special measurements of aright triangle(a triangle with oneanglemeasuring$90°$). Remember that the two sides of a right triangle which form the right angle are called thelegs, and the third side (opposite the right angle) is called thehypotenuse.

There are three basic trigonometric ratios:sine,cosine, andtangent. Given a right triangle, you can find the sine (or cosine, or tangent) of either of the non-$90°$angles.

$\begin{array}{l}\text{sine}=\frac{\text{length of the leg opposite to the angle}}{\text{length of hypotenuse}}\text{abbreviated "sin"}\\ \text{cosine}=\frac{\text{length of the leg adjacent to the angle}}{\text{length of hypotenuse}}\text{abbreviated "cos"}\\ \text{tangent}=\frac{\text{length of the leg opposite to the angle}}{\text{length of the leg adjacent to the angle}}\text{abbreviated "tan"}\end{array}$

The length of the leg opposite $\angle A$is$a$. The length of the leg adjacent to$\angle A$is$b$, and the length of the hypotenuse is$c$.

The sine of the angle is given by the ratio "opposite over hypotenuse." So,

$\sin \angle A=\frac{a}{c}$

The cosine is given by the ratio "adjacent over hypotenuse."

$\cos \angle A=\frac{b}{c}$

The tangent is given by the ratio "opposite over adjacent."

$\tan \angle A=\frac{a}{b}$

Generations of students have used the mnemonic "SOHCAHTOA" to remember which ratio is which. (Sine:Opposite overHypotenuse,Cosine:Adjacent overHypotenuse,Tangent:Opposite overAdjacent.)

Other Trigonometric Ratios

The other common trigonometric ratios are:

$\begin{array}{l}\text{secant}=\frac{\text{length of hypotenuse}}{\text{length of the leg adjacent to the angle}}\text{abbreviated "sec"}\sec \left(x\right)=\frac{1}{\cos \left(x\right)}\\ \text{cosecant}=\frac{\text{length of hypotenuse}}{\text{length of the leg opposite to the angle}}\text{abbreviated "csc"}\csc \left(x\right)=\frac{1}{\sin \left(x\right)}\\ \text{secant}=\frac{\text{length of the leg adjacent to the angle}}{\text{length of the leg opposite to the angle}}\text{abbreviated "cot"}\cot \left(x\right)=\frac{1}{\tan \left(x\right)}\end{array}$

The length of the leg opposite$\angle A$is$a$. The length of the leg adjacent to$\angle A$is$b$, and the length of the hypotenuse is$c$.

The secant of the angle is given by the ratio "hypotenuse over adjacent". So,

$\sec \angle A=\frac{c}{b}$

The cosecant is given by the ratio "hypotenuse over opposite".

$\csc \angle A=\frac{c}{a}$

The cotangent is given by the ratio "adjacent over opposite".

$\cot \angle A=\frac{b}{a}$