Basic Trigonometry: Sin Cos Tan (NancyPi) - YouTube

Hi guys! I'm Nancy.

And I'm going to show you how to use the basic trig functions

and how to find the values of the

six basic trigonometric functions.

So that is: sine, cosine, tangent

And their partners: cosecant, secant and cotangent.

So first we're going to look at just sin, cos and tan.

And we're going to use the memory trick SOH-CAH-TOA

which a lot of people say quickly as "sohcahtoa".

But this is what it is, it's a memory trick to remember

what sin, cos and tan mean. So we'll use that.

All of these 6 trig functions involve right triangles.

That is, a triangle that has a right angle in the corner.

Which is noted by the corner symbol down here.

So it's 90 degrees. A 'right angle'.

And you will also know where theta is.

It will be labeled on your triangle.

Theta is just an angle.

It's an angle that is not your right angle.

What's important to know is what each of the sides of the triangle are

in relation to theta. What I mean by that is...

The longest side on the triangle is called the 'hypotenuse'.

So you'll just want to remember that the longest side

is the hypotenuse.

The other side, that is next to theta, that is not the hypotenuse

is called the 'adjacent' side.

It's adjacent to theta, but it's not the hypotenuse.

And the other side, the third side, is opposite theta.

Directly opposite theta. And it's the 'opposite' side.

So notice that the hypotenuse, is immediately opposite the right angle.

The 'opposite side' is opposite theta.

And then the 'adjacent side' is the other side

next to theta that is not the hypotenuse.

So you will use those to evaluate sin, cos and tan.

So you might be working on a problem that looks something like this

where you're given a triangle. A right triangle.

With a right angle.

Some theta labeled on your diagram.

and then side lengths that are given to you.

So for instance you might be given

that the longest side, the diagonal, is 5.

This side length could be 3...

and this side 4.

And then the question might be:

Find sin of theta, cos of theta and tan of theta.

So let's find those.

sin of theta

sin(theta) is equal to: opposite/hypotenuse

So this memory trick, SOH-CAH-TOA,

can help you know what values to use for sin, cos and tan.

For Sin it's going to be Opposite / Hypotenuse. It's the 'SOH' part.

So sin equals opposite / hypotenuse.

So I'll write that out: Opposite over hypotenuse.

And in this particular problem, the opposite side is 4.

This is directly opposite theta. 4.

And the hypotenuse, the longest side, is 5. So put 5.

So your answer for sin(theta) is 4/5.

So in the same way you can find cos using your memory trick.

The acronym SOC-CAH-TOA. So cos is the 'CAH' part of this trick.

C-A-H. And that stands for Cos equals Adjacent / Hypotenuse.

cos = adjacent / hypotenuse

And so in this problem, the adjacent side to theta is 3.

And the hypotenuse is again 5.

So your cos value is 3/5.

And then finally if you have to find tan(theta)

just remember this part of the name SOH-CAH-TOA.

The TOA part.

T-O-A. So Tangent equals Opposite / Adjacent.

The opposite side in this triangle is 4

and the adjacent side is 3. So you have 4/3.

So your tan(theta) = 4/3.

OK. Say you actually need to find the values of csc, sec, or cot.

One of the other 3 basic trig functions.

The easiest way is to first find the value of their partner trig functions.

sin, cos or tan.

And then take the reciprocal. 1/sin. 1/cos. 1/tan.

So let me show you an example.

Say you wanted to find cosecant of theta. csc(theta).

First, find its partner, sin(theta).

And in this example sin(theta), remember is opposite/hypotenuse.

From SOH-CAH-TOA. sin is opposite/hypotenuse.

So in this triangle it is 4/5.

csc(theta) is by definition: 1/sin(theta)

And what that really means is that you can just

flip the numerator and denominator of sin. Flip top and bottom.

So instead of 4/5, it's 5/4 for csc(theta).

So your csc(theta) = 5/4.

Now if you wanted to find secant of theta, sec(theta),

you would first find cos(theta).

In this triangle, remember cos is adjacent/hypotenuse,

so here it's 3/5.

Then, to get secant theta, sec(theta),

you would take 1/cos(theta).

And that can be found just by flipping top and bottom.

And so instead of 3/5, 1 over that would turn out to be 5/3.

The reciprocal.

So we have sec(theta) is 5/3.

And then finally if you wanted to find cot(theta)

you would first find tan(theta).

So tan(theta) is opposite/adjacent.

So in this triangle it's 4/3

and then to get cot

you would do 1/tan.

Which is the reciprocal of this value. So it's 3/4.

So you get: cot(theta) = 3/4.

That is the fastest, easiest way to find csc, sec and cot.

OK. I want to show you some things that can trip people up.

One thing that confuses people is when

the right triangle is drawn at a different orientation.

So if it's sideways or on its hypotenuse.

So for instance you could have a triangle

that's drawn like this.

Still a right triangle, but looks kinda upside down.

Your theta could be here.

You could have a triangle that looks like...

this.

So it's the reverse of the one in our example.

You could have...

a triangle drawn...

with your theta up here. Instead of down here.

And the one that's most confusing to people

is when the right triangle is drawn

so that its hypotenuse is horizontal.

It's sort of on the ground.

So here's an example of that.

If your right angle was actually up here

then your hypotenuse turned out to be horizontal

and your theta could be here or here.

So let's look at these examples

and label where opposite, adjacent and hypotenuse are

for these triangles.

In this example, in all of these triangles actually,

you can rely on your hypotenuse being the longest side.

So in this triangle

you can still tell that this is the longest side

so it would be your hypotenuse.

The other side that's next to theta

is still your adjacent side.

So this would be your adjacent.

And then, the other side that is opposite theta

directly across from theta is opposite side.

Let's label these are well.

Across from theta is your opposite side.

longest side is your hypotenuse

and the other side next to theta is your adjacent.

Over here, your adjacent is this side.

Your opposite is down here.

And your hypotenuse is the longest side.

OK. Down here. Longest side is still your hypotenuse.

The one that's opposite the right angle.

The one opposite theta is your opposite side.

And then the other side next to theta is your adjacent.

There's one more thing that trips people up in these problems

when your finding the value of trig functions.

When you write your final answer

make sure you put theta.

You don't want to write something like this

that your "sin = 3/5".

You want to write sin of an angle.

sin(theta) = a number.

If you don't write the angle, this doesn't have any meaning.

I know a lot of people forget it, I get it. I know.

It's a lot of remember and these trig functions

are all new and weird, but

if you don't write the angle it doesn't really make sense.

It doesn't really have any meaning.

So make sure you write your full answer as

sin(of an angle), cos(theta), tan(theta) or tan(x).

Whatever your angle is called.

I hope that helped you understand

sines, cosines, tangents,

cosecant, secant, cotangents!

I know trig functions are super fun.

It's OK.

You don't have to like math, but...

but you can like my video!

So if this helped you, please click like or subscribe below!