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## Geometry (all content)

### Course: Geometry (all content)>Unit 13

Lesson 2: Introduction to the trigonometric ratios

# Intro to the trigonometric ratios

Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. Sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. They are often written as sin(x), cos(x), and tan(x), where x is an angle in radians or degrees. Created by Sal Khan.

## Want to join the conversation?

• What do you use for the adjacent side if you're trying to figure out the sin cos and tan for the right angle? •   Well, in beginning trigonometry, it's convenient to evaluate sin/cos/tan by using soh-cah-toa, but later, as you get into the unit circle and you start taking taking stuff like sin(135) and tan(-45) you don't use the adjacent-opposite-hypotenuse much anymore. If you can think of it intuitively, though, sin(90) means that the opposite side is infinitely long, and the hypotenuse is also infinitely long, so sin(90)=1. cos(90) means adjacent over the hypotenuse, which is infinitely long given that the angle is 90 degrees, so any number over infinity is 0, so cos(90)=0. tan(90)=sin(90)/cos(90)=1/0, so tan(90) doesn't exist.
• How can you figure out which is the opposite or the adjacent? •   The opposite side is the side opposite of the angle that you are trying to solve for. The adjacent side is the side next to the angle you are solving for.
• A small question. We name the ratios as "Sine", "Cosine", "Tangent", "Cotagent", "Secant" and "Cosecant". The last four can be drawn of circle. Is "Sine" also a part of circle. I mean can it be drawn on circle like tangent and secant. I know its a useless question, but I was just wondering. Thanks for your time. •   I think that's a great question! This is a pretty cool story (to me at least). The word that the Arabs used for sine was the same as their word for "chord", but when a European translated it into Latin he read it wrong and translated it as sinus, which is the Latin word for chest. So it's a historical accident that secant and tangent have geometric meanings but sine doesn't.
• Why does Sal (the person talking in the video) use theta or some other greek letter for the angles instead of a normal variable, like x or y, for every angle he shows the sin, cos, and tan for? •  The reason is because in the world of math (not khan academy's "world of math"), mathematicians usually use x and y for missing lengths, and use Greek letters for unknown angles, most likely in honor of Elucid, founder of geometry, who was Greek.

Hope that helps!
• My question is around how to calculate sin, cos or tan. I've pushed the sin/cos/tan button many times on my calculator with no _idea_ what is actually happening. From doing some of my own research, it seems like a Taylor Series may have to be used? Is this the only way? Did someone once sit down and measure every angle and every side of the triangle to get each ratio into a large table? Would it then be something like a look up table with the calculator simply searching for the closest ratio that matches what is typed into the calculator? • sin cos and tan are basically just functions that relate an angle with a ratio of two sides in a right triangle. Sin is equal to the side opposite the angle that you are conducting the functions on over the hypotenuse which is the longest side in the triangle. Cos is adjacent over hypotenuse. And tan is opposite over adjacent, which means tan is sin/cos. this can be proved with some basic algebra.
• To help you guys understand SOH CAH TOA even better, I decided to create this comment as a question.

SOH: [S is Sine, O is Opposite, H is Hypotenuse]. Then, [Sine= Opposite/Hypotenuse].

CAH: [C is Cosine, A is Adjacent, H is Hypotenuse]. Then, [Cosine= Adjacent/Hypotenuse].

TOA: [T is Tangent, O is Opposite, A is Adjacent]. Then, [Tangent= Opposite/Adjacent].

To simplify it to make you guys understand even better, knowing the short form for it, I shall show it down below.

SOH: S= O/H

CAH: C= A/H

TOA: T= O/A

Now, since I already told you guys about the SOH CAH TOA form, I shall give you guys an example.

For Example:
If you are given a triangle where the two significantly shorter sides are given and you wish to obtain the longest side termed the hypotenuse, you recognize that it satisfies the terms necessary to use either the SOH or CAH form; You, therefore, proceed to identify which is adjacent and which is the opposite. The opposite, which is clearly identifiable due to its name, is the side that is directly OPPOSITE the given angle. The adjacent is therefore the side which forms a 90° angle to the opposite.

Another Example:
For purposes, the given angle is 45°. So you then proceed to imply due to the SOH form that Sine45= opposite divided by the hypotenuse. For you to obtain the hypotenuse, we transpose it for the hypotenuse to become the subject of the formula. By cross multiplying we obtain the formula:
Hypotenuse = Opposite divided by Sine45.

Note: Some of these contents are copied from @machyl69.

Hope This Helps,
Thank You! • At
what do you call the hypotenuse if it's not a right angle triangle • How do you use trigonometry on 3d and even 4d shapes and objects?   