The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are defined for acute angle $A$ below:
[[☃ image 2]]
In these definitions, the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.
SOH-CAH-TOA: an easy way to remember trig ratios
The word sohcahtoa helps us remember the definitions of sine, cosine, and tangent. Here's how it works:
Acronym Part | Verbal Description | Mathematical Definition
- | - | -
$\Large S\blueD{O}\purpleC{H}$ | $\text{S}$ine is $\text{\blueD{O}}$pposite over $\text{\purpleC{H}}$ypotenuse | $\sin(A) = \dfrac{\text{\blueD{Opposite}}}{\text{\purpleC{Hypotenuse}}} $
$\Large C\maroonC{A}\purpleC{H}$ | $\text{C}$osine is $\text{\maroonC{A}}$djacent over $\text{\purpleC{H}}$ypotenuse |$ \cos(A) = \dfrac{\text{\maroonC{Adjacent}}}{\text{\purpleC{Hypotenuse}}} $
$\Large T\blueD{O}\maroonC{A}$ | $\text{T}$angent is $\text{\blueD{O}}$pposite over $\text{\maroonC{A}}$djacent | $ \tan(A) = \dfrac{\text{\blueD{Opposite}}}{\text{\maroonC{Adjacent}}} $
For example, if we want to recall the definition of the sine, we reference $S\blueD{O}\purpleC{H}$, since sine starts with the letter S. The $\text{\blueD{O}}$ and the $\text{\purpleC{H}}$ help us to remember that sine is $\text{\blueD{opposite}}$ over $\text{\purpleC{hypotenuse}}$!
Example
Suppose we wanted to find $\sin( A)$ in $\triangle ABC$ given below:
[[☃ image 1]]
Sine is defined as the ratio of the $\text{\blueD{opposite}}$ to the $\text{\purpleC{hypotenuse}}$ $(S\blueD{O}\purpleC{H}) $. Therefore:
[[☃ image 1]]
$\begin{align}\sin( A)&=\dfrac{\blueD{\text{ opposite }} }{ \purpleC{\text{ hypotenuse}} }\\
&=\dfrac{\blueD{BC}}{\purpleC{AB}}\\\
&=\dfrac{\blueD{3}}{\purpleC{5}} \\\
\end{align}$
Here's another example in which Sal walks through a similar problem:
[[☃ video 1]]
Practice
Triangle 1: $\triangle DEF$
[[☃ image 4]]
[[☃ graded-group 1]]
[[☃ graded-group 2]]
[[☃ graded-group 3]]
Triangle 2: $\triangle GHI$
[[☃ image 3]]
[[☃ graded-group 3]]
[[☃ graded-group 4]]
[[☃ graded-group 5]]
[[☃ graded-group 3]]