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## Geometry (all content)

# Trigonometric ratios in right triangles

CCSS.Math: ,

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: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 |
---|---|---|

S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff | start 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 textypotenuse | sine, 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 #aa87ff | start 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 textypotenuse | cosine, 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 #ed5fa6 | start 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 textdjacent | tangent, 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:

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:

Here's another example in which Sal walks through a similar problem:

## Practice

**Triangle 1: triangle, D, E, F**

**Triangle 2: triangle, G, H, I**

## Want to join the conversation?

- hey I have a question

what if we have a triangle with no known sides but 2 angles(including one right angle) is given then how will we find the 3rd angle and 3 sides? is it possible?(18 votes)- If you know two angles of a triangle, it is easy to find the third one. Since the three interior angles of a triangle add up to 180 degrees you can always calculate the third angle like this:

Let's suppose that you know a triangle has angles 90 and 50 and you want to know the third angle. Let's call the unknown angle x.

x + 90 + 50 = 180

x + 140 = 180

x = 180 - 140

x = 40

As for the side lengths of the triangle, you need more information to figure those out. A triangle of side lengths 10, 14, and 9 has the same angles as a triangle with side lengths of 20, 28, and 18.(7 votes)

- How is theta defined in accurate mathematical language?(6 votes)
- Theta is a variable used to represent an unknown angle.(24 votes)

- What is the etymology of sin, cos and tan?(9 votes)
**From Wikipedia - Trigonometric Functions - Etymology**

The word sine derives from Latin*sinus*, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word*jaib*, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. The choice was based on a misreading of the Arabic written form*j-y-b*(جيب), which itself originated as a transliteration from Sanskrit*jīvā*, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".

The word tangent comes from Latin*tangens*meaning "touching", since the line*touches*the circle of unit radius, whereas*secant*stems from Latin*secans*—"cutting"—since the line*cuts*the circle.

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the*cosinus*as an abbreviation for the*sinus complementi*(sine of the complementary angle) and proceeds to define the*cotangens*similarly.(27 votes)

- IS there ANY way to easily remember the SIN, COS and TAN formulas?? Any tips and tricks?(1 vote)
- SOH CAH TOA. The sine of theta, θ, or sine(θ = opposite side divided by hypotenuse, cosine(θ = adjacent side divided by hypotenuse, and tangent(θ = opposite divided by adjacent side. Or SOH CAH TOA(4 votes)

- How to find the sin, cos and tan of the 90 degree angle? Will we follow the same procedure as we did with the other two angles?(5 votes)
- If we consider the right angle, the side opposite is also the hypotenuse. So sin(90)=h/h=1.

By pythagorean theorem, we get that sin^2(90)+cos^2(90)=1. So, substituting, 1+cos^2(90)=1

cos^2(90)=0

cos(90)=0

And we see that tan(90)=sin(90)/cos(90)=1/0. So tan(90) is undefined.(7 votes)

- Based on the first paragraph, "The ratios of the sides of a right triangle are called trigonometric ratios.", if in trigonometry the ratios of the sides of a triangle are called 'trigonometric ratios' then what if the triangle is not a right triangle. Will the ratios of the sides of that triangle have a different label. And based on my question, how will the mnemonic 'soh cah toa' help find the sides of the 'non- right triangle' triangle? Are there more methods to find the sides of a triangle relative to trigonometric functions or formula?(3 votes)
- Good questions, it's clear you are thinking about where this is going. The laws of sines and cosines can be used to help you figure out the relationships of the sides and angles for triangles that are not right triangles. There are some great videos:

https://www.khanacademy.org/math/geometry/hs-geo-trig/modal/v/law-of-sines

https://www.khanacademy.org/math/geometry/hs-geo-trig/modal/v/law-of-cosines-example

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solving-general-triangles/v/law-of-cosines-word-problem(10 votes)

- I've heard that there are other trigonometric functions out there, with names like versine. Who decided that sine, cosine, and tangent would be the ones we learn in school? What happened to the others?(3 votes)
- I would guess that it's because these functions are technically more complex than the ones we learn in school. For example, versine(x) = 1 - cos(x). Applications of these functions seem to be applicable to navigation, especially across a spherical plane. However, with the progression of technology (I assume) these older functions have grown less practical and have fallen away in favor of manipulations of the more familiar 6 trig functions we study today.(6 votes)

- why is sin, cos and tan change?(2 votes)
- sin cos and tan changes based on the angle you choose.it is all matter of perspective.(9 votes)

- Can you find the sin, cos and tan of an more than 90 degree angle(4 votes)
- Yes. If you look at the sine, cosine and tangent graphs, you'll see that they go on forever. The sine and cosine oscillate between -1 and 1 and the tangent function has a range of -infinity to infinity, and repeats every 180°.(5 votes)

- Did anyone else notice that Sal said "Sine is opposite over adjacent" instead of "Sine is opposite over hypotenuse" about the time10:30?(4 votes)
- Yes, I think that is a mistake. I think they may have mentioned it, but its always good to see those types of errors.(2 votes)