Hypotenuse, opposite, and adjacent (article)

Use Tangent to Determine The Length of an Adjacent Side of a Right Triangle
Use Tangent to Determine The Length of an Adjacent Side of a Right Triangle

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High school geometry

Hypotenuse, opposite, and adjacent

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 :

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 :

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 , 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(36 votes)
    • 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.(73 votes)
  • 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.(20 votes)
    • 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!!(43 votes)
  • why is trigonometry important?(11 votes)
    • 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.(30 votes)
  • why do we need to learn trigonometry?why are they important?where did the names sine cos tan come from?(13 votes)
    • 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.(27 votes)
  • Is trigonometry just about triangle?(9 votes)
    • Yes the roots come from tri (three) gono (angle) metry (measure)
      see http://mathforum.org/library/drmath/view/52595.html(17 votes)
  • why do we need to learn trigonometry?(3 votes)
    • Trigonometry is part of the standard high school curriculum, but it’s not an essential subject for nothing. Many career choices involve studying trigonometry, especially STEM fields such as science, engineering, or technology. In the end, it depends on you and your career choice. Because, if anything, trigonometry is very useful for learning physics and astronomy. In some cases, you may have to use trigonometry for videogame design and programming as well. Moreover, at least in college you may have to learn trigonometry as a sort of gatekeeper before you can take core curriculum classes. This may be the case for computer science where students have to learn calculus or even physical therapy. To summarize, you don’t have to love trigonometry, but many of the world’s most valuable inventions and discoveries couldn’t have existed without the math of tirangles, and the same thing applies to the future you want to pursue, whether academic success or honorable achievements.
      (Fun Fact; man wouldn’t have landed on the moon if people didn’t know/study trigonometry👀)
      It’s an interesting math, and though it can be hard, it’s the foundation of many amazing works today, but of course you’re free to like the subject or dislike it, I’m just here to spread the benefit!🙌
      Hope I helped!🧡(28 votes)
  • Where is it used in real life(7 votes)
    • The GPS satellite system to tell where you are, surveying, building, anime, etc.(15 votes)
  • How do you know which one is the opposite and the adjacent side?(5 votes)
    • 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.(8 votes)
  • my hungry self could not do these math problems(4 votes)
    • then eat food(16 votes)
  • What is cosine and sine?(3 votes)
    • Cosine equals to the adjacent divided by the hypotenuse, and sine equals to the opposite divided by the hypotenuse. Think of this: soh, cah, and toa. soh means sine = opposite/hypotenuse, cah means cosine = adjacent/hypotenuse, and toa means tangent = opposite/adjacent.
      Hope that helps.(14 votes)

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