Basic Trigonometry Introduction to trigonometry
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- In this video I wanna give you the
- basics of Trigonometry.
- It's sounds like a very complicated topic
- but you're gonna see this is just the study
- of the ratios of sides of Triangles.
- The "Trig" part of "Trigonometry" literally means
- Triangle and the "metry" part literally means
- Measure. So let me just give you some examples here.
- I think it'll make everything pretty clear.
- So let me draw some right triangles, let me just draw
- one right triangle. So this is a right triangle.
- When I say it's a right triangle, it's because
- one of the angles here is 90 degrees.
- This right here is a right angle.
- It is equal to 90 degrees.
- And we will talk about other ways
- to show the magnitude of angles in future videos.
- So we have a 90 degree angle.
- It's a right triangle, let me put some
- lengths to the sides here. So this side over here is maybe 3. This height right over there is 3.
- Maybe the base of the triangle right over here is 4.
- and then the hypotenuse of the triangle over here is 5.
- You only have a hypotenuse when you have a right triangle.
- It is the side opposite the right angle and it is the longest side of a right triangle.
- So that right there is the hypotenuse.
- You've probably learned that already from geometry.
- And you can verify that this right triangle - the sides work out -
- we know from the Pythagorean theorem, that 3 squared
- plus 4 squared, has got to be equal to the length of the longest side,
- the length of the hypotenuse squared is equal to 5 squared
- so you can verify that this works out
- that this satisfies the Pythagorean theorem.
- Now with that out of the way let's learn a little bit of Trigonometry.
- The core functions of trigonometry,
- we're going to learn a little more about what these functions mean.
- There is the sine, the sine function.
- There is the cosine function, and there is the tangent function.
- And you write sin, or S-I-N, C-O-S, and "tan" for short.
- And these really just specify, for any angle in this triangle,
- it will specify the ratios of certain sides.
- So let me just write something out.
- This is really something of a mnemonic here,
- so something just to help you remember the definitions of these functions,
- but I'm going to write down something called "soh cah
- toa", you'll be amazed how far this mnemonic will take you in trigonometry.
- We have "soh cah toa", and what this tells us is;
- "soh" tells us that "sine" is equal to opposite over hypotenuse.
- It's telling us. And this won't make a lot of sense just now,
- I'll do it in a little more detail in a second.
- And then cosine is equal to adjacent over hypotenuse.
- And then you finally have tangent,
- tangent is equal to opposite over adjacent.
- So you're probably saying, "hey, Sal, what is all this "opposite"
- "hypotenuse", "adjacent", what are we talking about?"
- Well, let's take an angle here.
- Let's say that this angle right over here is theta,
- between the side of the length 4, and the side
- of length 5. This is theta.
- So lets figure out the sine of theta,
- the cosine of theta, and what the tangent of
- theta are.
- So if we first want to focus on the sine of theta,
- we just have to remember "soh cah toa",
- sine is opposit over hypotonuse, so sine of theta is equal to the opposite -
- so what is the opposite side to the angle?
- So this is our angle right here, the opposite side,
- if we just go to the opposite side,
- not one of the sides that are kind of adjacent to the angle,
- the opposite side is the 3,
- if you're just kinda - it's opening on to that 3,
- so the opposite side is 3.
- And then what is the hypotenuse?
- Well, we already know - the hypotenuse here is 5.
- So it's 3 over 5.
- The sine of theta is 3/5.
- And I'm going to show you in a second, that the sine of theta -
- if this angle is a certain angle - it's always going to be 3/5.
- The ratio of the opposite to the hypotenuse is always going to be the same,
- even if the actual triangle were a larger triangle
- or a smaller one.
- So I'll show you that in a second.
- So let's go throught all of the trig functions.
- Let's think about what the cosine of theta is.
- Cosine is adjacent over hypotenuse, so remember -
- let me label them.
- We already figured out that the 3 was the opposite side.
- This is the opposite side.
- And only when we're talking about this angle.
- When we're talking about this angle - this side is opposite to it.
- When we're talking about this angle, this 4 side
- is adjacent to it,
- it's one of the sides that kind of make up - that
- kind of form the vertex here.
- So this right here is the adjacent side.
- And I want to be very clear,
- this only applies to this angle.
- If we're talking about that angle,
- then this green side would be opposite,
- and this yellow side would be adjacent.
- But we're just focusing on this angle right over here.
- So cosine of this angle - so the adjacent side of this angle is 4,
- so the adjacent over the hypotenuse,
- the adjacent, which is 4, over the hypotenuse,
- 4 over 5.
- Now let's do the tangent.
- Let's do the tangent.
- The tangent of theta: opposite over adjacent.
- The opposite side is 3. What is the adjacent side?
- We've already figured that out, the adjacent
- side is 4.
- So knwoing the sides of this right triangle,
- we were able to figure out the major trig ratios.
- And we'll see that there are other trig ratios,
- but they can all be derived from these three
- basic trig functions.
- Now, let's think about another angle in this triangle,
- and I'll re-draw it, because my triangle is getting a little bit messy.
- So I'll re-draw the exact same triangle.
- The exact same triangle.
- And, once again, the lengths of this triangle are -
- we have length 4 there, we have length 3 there,
- we have length 5 there.
- In the last example we used this theta.
- But let's do another angle, let's do another angle up here,
- and let's call this angle - I don't know, I'll think of something,
- a random Greek letter.
- So let's say it's psi.
- It's, I know, a little bit bizarre.
- Theta is what you normally use,
- but since I've already used theta, let's use psi.
- Or actually - let me simplify it,
- let me call this angle x.
- Let's call that angle x.
- So let's figure out the trig functions for that angle x.
- So we have sine of x, is going to be equal to what?
- Well sine is opposite over hypotenuse.
- So what side is opposite to x?
- Well it opens on to this 4,
- it opens on to the 4.
- So in this context, this is now the opposite,
- this is now the opposite side.
- Remember: 4 was adjacent to this theta,
- but it's opposite to x.
- So it's going to be 4 over -
- now what's the hypotenuse?
- Well, the hypotenuse is going to be the same
- regardless of which angle you pick,
- so the hypotenuse is now going to be 5,
- so it's 4/5.
- Now let's do another one; what is the cosine of x?
- So cosine is adjacent over hypotenuse.
- What side is adjacent to x, that's not the hypotenuse?
- You have the hypotenuse here.
- Well the 3 side, it's one of the sides that
- forms the vertex that x is at, that's not the hypotenuse,
- so this is the adjacent side.
- That is the adjacent.
- So it's 3 over the hypotenuse,
- the hypotenuse is 5.
- And then finally, the tangent.
- We want to figure out the tangent of x.
- Tangent is opposite over adjacent,
- "soh cah toa", tangent is opposite over adjacent,
- opposite over adjacent.
- The opposite side is 4.
- I want to do it in that blue color.
- The opposite side is 4, and the adjacent side is 3.
- And we're done!
- And in the next video I'll do a ton of more examples of this,
- just so that we really get a feel for it.
- But I'll leave you thinking of what happens when
- these angle start to approach 90 degrees,
- or how could they even get larger than 90 degrees.
- And we'll see that this definition,
- the "soh cah toa" definition takes us a long way
- for angles that are between 0 and 90 degrees,
- or that are less than 90 degrees.
- But they kind of start to mess up
- really at the boundries.
- And we're going to introduce a new definition,
- that's kind of derived from the "soh cah toa" definition
- for finding the sine, cosine and tangent
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