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## Trigonometry

### Course: Trigonometry>Unit 1

Lesson 2: Introduction to the trigonometric ratios

# Trigonometric ratios in right triangles

Learn how to find the sine, cosine, and tangent of angles in right triangles.
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:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is labeled opposide. Side A C is labeled adjacent. Side A B is labeled hypotenuse.
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 PartVerbal DescriptionMathematical Definition
S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ffstart text, S, end textine is start text, start color #11accd, O, end color #11accd, end textpposite over start text, start color #aa87ff, H, end color #aa87ff, end textypotenusesine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction
C, start color #ed5fa6, A, end color #ed5fa6, start color #aa87ff, H, end color #aa87ffstart text, C, end textosine is start text, start color #ed5fa6, A, end color #ed5fa6, end textdjacent over start text, start color #aa87ff, H, end color #aa87ff, end textypotenusecosine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction
T, start color #11accd, O, end color #11accd, start color #ed5fa6, A, end color #ed5fa6start text, T, end textangent is start text, start color #11accd, O, end color #11accd, end textpposite over start text, start color #ed5fa6, A, end color #ed5fa6, end textdjacenttangent, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, end fraction
For example, if we want to recall the definition of the sine, we reference S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, since sine starts with the letter S. The start text, start color #11accd, O, end color #11accd, end text and the start text, start color #aa87ff, H, end color #aa87ff, end text help us to remember that sine is start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text over start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text!

## Example

Suppose we wanted to find sine, left parenthesis, A, right parenthesis in triangle, A, B, C given below:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units.
Sine is defined as the ratio of the start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text to the start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text left parenthesis, S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, right parenthesis. Therefore:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units. Sides A B and B C are highlighted.
\begin{aligned}\sin( A)&=\dfrac{\blueD{\text{ opposite }} }{ \purpleC{\text{ hypotenuse}} }\\\\ &=\dfrac{\blueD{BC}}{\purpleC{AB}}\\\\\\ &=\dfrac{\blueD{3}}{\purpleC{5}} \\\\\\ \end{aligned}
Here's another example in which Sal walks through a similar problem:
Trigonometric ratios in right trianglesSee video transcript

## Practice

Triangle 1: triangle, D, E, F
Triangle D E F with angle E D F being ninety degrees. Side D F is twelve units. Side E F is thirteen units. Side D F is five units.
cosine, left parenthesis, F, right parenthesis, equals

sine, left parenthesis, F, right parenthesis, equals

tangent, left parenthesis, F, right parenthesis, equals

Triangle 2: triangle, G, H, I
Triangle G H I with angle G I H being ninety degrees. Side H I is fifteen units. Side I G is eight units. Side H G is seventeen units.
cosine, left parenthesis, G, right parenthesis, equals

sine, left parenthesis, G, right parenthesis, equals

tangent, left parenthesis, G, right parenthesis, equals

Challenge problem
In the triangle below, which of the following is equal to start fraction, a, divided by, c, end fraction?
A right triangle with a ninety-degree angle, a twenty-degree angle, and seventy-degree angle. The side opposide of the twenty-degree angle is a units. The side opposite of seventy-degree angle is b units. The side opposite of the ninety-degree angle is c units.