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## High school geometry

### Course: High school geometry>Unit 5

Lesson 4: Ratios in right triangles

In a right triangle, the hypotenuse is the longest side, an "opposite" side is the one across from a given angle, and an "adjacent" side is next to a given angle.
We use special words to describe the sides of right triangles.
The hypotenuse of a right triangle is always the side opposite the right angle. It is the longest side in a right triangle.
A right triange A B C where Angle C is ninety degrees. Inside the triangle, an arrow points from point C to the hypotenuse. The hypotenuse is labeled hypotenuse.
The other two sides are called the opposite and adjacent sides. These sides are labeled in relation to an angle.
The opposite side is across from a given angle.
A right triange A B C where Angle C is ninety degrees. Inside the triangle, an arrow points from point A to side B C. Side B C is labeled opposite.
The adjacent side is the non-hypotenuse side that is next to a given angle.
A right triange A B C where Angle C is ninety degrees. Inside the triangle, an arrow points from point A to side A C. Side A C is labeled adjacent.
Putting it all together from the perspective of angle, A:
A right triange A B C where Angle C is ninety degrees. Side A B is labeled hypotenuse. Side B C is labeled opposite. Side A C is labeled adjacent. The angle of reference is at angle A.
And from angle, B:
A right triange A B C where Angle C is ninety degrees. Side A B is labeled hypotenuse. Side A C is labeled opposite. Side B C is labeled adjacent. The angle of reference is at angle B.

## Practice

Problem 1
• Current
Relative to angle G, which side is the adjacent side?
A right triange E G M. The short leg is E M. The long leg is M G. The longest side is G E. The angle of reference is at angle G.

## Why are these words important?

We're about to learn the trigonometric functions—sine, cosine, and tangent—which are defined using the words hypotenuse, opposite, and adjacent.

## Want to join the conversation?

• who is the largest and the shortest of these three words hypotenuse opposite and adjacent
• The shortest side is the one opposite the smallest angle. If the angle you already know is the shortest one, then the shortest side is opposite it. However, if the angle you already know is the medium one, then the shortest side is adjacent to it.

The hypotenuse is always the longest side in a right triangle because it is opposite of the largest angle, the ninety degree angle.
• Can any of the calculations of trigonometry be applied to non-right triangles? Seems like a very niche area if it only covers right triangles.
• They sure can! You learn about the unit circle in Precalculus! The unit circle is far more complicated than right triangle trig though, you might want to wait a while before learning it. :)
Happy holidays!!
• why is trigonometry important?
• Trigonometry is about understanding triangles, and every other polygon can be disassembled into triangles. So trigonometry becomes an important aspect of all of plane geometry.
• why do we need to learn trigonometry?why are they important?where did the names sine cos tan come from?
• Trigonometry is very useful in any type of physics, engineering, meteorology, navigation, etc... (Wherever geometry is useful, trig is almost certain to also be useful). Trig isn't for everyone, however if little billy wants to calculate how tall a building is without producing the world's longest tape measure, he's gonna need some trig. The name sine (from what i know) comes from the latin word sinus, meaning hole or cavity, basically translation after translation of the word we ended with hole, which turned into sinus, sine for short (I may be wrong, but that is what I remember). The name cosine comes from the fact that sine and cosine are co-functions, (due to the fact that sin(x-90)=cosx. Tangent is not as easy to explain, it has to do with geometry and tangent lines.
• Is trigonometry just about triangle?
• Where is it used in real life
• The GPS satellite system to tell where you are, surveying, building, anime, etc.
• How do you know which one is the opposite and the adjacent side?
• The problem will say, "relative to angle ___." Connect that angle to the right angle in the triangle, and that's the adjacent side. Then you know the hypotenuse(opposite of the right angle) and the adjacent side, so the only other side must be the opposite side.
• Why is trigonometry associated with right angled triangles? Why not equilateral, obtuse and acute?
• trigonometry does not only involve right angle triangles it involves all types of triangles,
use of rules such as the sine rule and the cosine rules are applicable
sine rule; (a/sinA)=(b/sinB)=(c/sinC)
cosine rule; a^2= b^2+c^2 - 2bccosA

Well there is a tangent rule...
(a-b)/(a+b)=(tan(a-b/2))/(tan(a+b/2))
• what are the applications of trigonometry in general life?
• Accurate calculation of distance between points, (if you ever hear the phrase "triangulate their position", that's what's going on!).

Sine waves and their equations are used for countless things, most anything that behaves like a wave such as sound, light, radio, etc.

Even things like game design, I was working on a spaceship game where I needed to have the ship turn and move on a 2D plane. Trig is how that problem is solved.
• isn't this concept important in the Pythagorean theorem