- anonymous

Will anybody explain to me how Sine Cosine and Tangent works? I missed school when they taught us, and when they tried to explain it to me I became brutally lost. :)

- katieb

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- zepdrix

Our trigonometric functions allow us to relate `the angle` of a triangle to its `sides`.

- zepdrix

We have this clever acronym for remembering the relationships:\[\Large\rm \color{red}{\text{Soh}}\color{green}{\text{Cah}}\color{royalblue}{\text{Toa}}\]
The `sine` of an angle is equivalent to the ratio of the `opposite` side to the `hypotenuse`.
That's what the `o` and `h` stand for.\[\large\rm \color{red}{\sin x=\frac{opposite}{hypotenuse}}\]

- zepdrix

Let's first look at a right triangle,
and make sure we understand how to label the sides.

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## More answers

- zepdrix

|dw:1444603738782:dw|So if this is my triangle, with angle x labeled here.
Do you know which side to label as your `hypotenuse`?

- anonymous

The side that is on top of the x, aka the "long side"

- zepdrix

Good. The longest side.
Another way I like to think of it, is in relation to the `right angle`.
It's always the side `opposite the right angle`.|dw:1444604392651:dw|

- zepdrix

Hmm with that idea in mind...
Which side do you think we would label as being located `opposite angle x`?

- anonymous

The side that is vertical. all the way to the right.

- zepdrix

|dw:1444604529049:dw|

- zepdrix

|dw:1444604552006:dw|Ok great.
So the last side we label as being `adjacent` or `next to` or angle x.

- zepdrix

|dw:1444604621344:dw|So if I have a particular triangle like this one,
based on our definition of sin x,
do you have an idea of how to find it using these values? :)

- anonymous

Inverse Cosine?

- zepdrix

No no, you're getting too fancy.
Using something like Inverse Sine would allow us to figure out our angle x, yes.
But we weren't up to that point yet XD
We just want \(\large\rm \sin x=?\)

- anonymous

I thought you said to find the angle :P so we want sin(x) which sine is Soa so Opposite over Adjacent?

- zepdrix

Woooops :O\[\Large\rm \color{red}{\text{Soh}}\color{green}{\text{Cah}}\color{royalblue}{\text{Toa}}\]Soh, not Soa ya silly billy >.<

- anonymous

Oh my lord x_x How did i even pass math class... SO! It'd be Opposite over Hypotenuse, yes?

- zepdrix

Ok good.\[\large\rm \color{red}{\sin x=\frac{opposite}{hypotenuse}}\]Based on the way we labeled these sides, it looks like we're using the 4 and 5, ya?\[\large\rm \color{red}{\sin x=\frac{4}{5}}\]

- anonymous

Yes yes. That's what the triangle tells us.

- zepdrix

And ya, if we wanted to solve for the angle, we could apply the inverse sine function,\[\large\rm \arcsin\left(\sin x\right)=\arcsin\left(\frac{4}{5}\right)\]On the left you the composition of a function and it's inverse,
which gives us back the argument,\[\large\rm x=\arcsin\left(\frac{4}{5}\right)\]

- zepdrix

|dw:1444605278799:dw|If that's too confusing,
another way to think of it is...
When you change from sine to inverse sine,
you just switch the stuff,

- anonymous

.O. ohh my... but one question... So i was told if you do normal sine, and you go to find the hypotenuse, your equasion would look something like this: 5/sin(x)= adjacent? Why is this?

- anonymous

*find the adjacent

- zepdrix

|dw:1444605485213:dw|Ok let's see if we can do something with this triangle.

- anonymous

*dies* x_x okayyy umm... SOH... 4/sin(30degrees) which will give us the adjacent...

- anonymous

Opposite. it gives us the opposite.

- zepdrix

Yah let's ignore the adjacent for now :)
If we want to solve for \(\large y\),the opposite side, Then we want to pay attention to THIS stuff,|dw:1444605655180:dw|

- anonymous

Yes. Sin(30)= y/4

- zepdrix

Mmm k good.
And how would you `isolate` the y?
You want to solve for y, so you need to get it alone somehow.
It's being `divided` by 4 right now.
How can we undo that?

- anonymous

multiply it by 4. which means we have to do it to the other side so 4times sin(30) = y

- zepdrix

Good.
And keep in mind that this 30 is like... locked in the sine function,
he can't interact with the 4 in any way.\[\large\rm 4\sin(30)\ne \sin(4\cdot30)\]

- zepdrix

\[\large\rm y=4\sin(30)\]And you would just use your calculator to finish that one off,
unless of course you've learned about your 30/60/90 triangle,
in which case you might be able to do it without a calculator.

- anonymous

So then we'd do sin(30) first. then multiply the answer by 4?

- anonymous

I got 2 when i plugged it into my calculator.

- zepdrix

Yay good job!
How would solve for the adjacent side, x?
Let's pretend that we don't know what y is.
So we can make sure of our cosine definition.

- zepdrix

|dw:1444606171089:dw|Again, we don't want more variables than x.
So we're dealing with these three pieces of information, ya?

- anonymous

Alright... so CAH... Cos(30)= x/4

- anonymous

and with that we'd need x alone so we multiply it by both sides correct?

- anonymous

4 is it XD oops.

- zepdrix

\[\large\rm \cos(30)=\frac{x}{4}\]Multiply 4? Ok seems like a good idea:\[\large\rm 4\cos(30)=\frac{x}{\cancel4}\cdot\cancel4\]\[\large\rm 4\cos(30)=x\]And unfortunately, this one is going to work out to a weird decimal length, but that's ok.

- anonymous

I got 3.46 for x o.o

- zepdrix

Here is a nice way to check your work:
Recall that your hypotenuse should be the `longest side`.
3.46 is shorter than 4 \(\large\rm \color{green}{\checkmark}\)
I tried to also draw that angle somewhat accurately to be a 30 degree angle.
So if that vertical length is 2,
would 3.5 be about right for the bottom length? yaaa that's prolly right!

- anonymous

Seems easy enough to remember.

- zepdrix

|dw:1444606546042:dw|Let's try this problem.
We need to find the length of the hypotenuse, z.

- anonymous

Cos(30)= 5/z

- zepdrix

\[\large\rm \cos(30)=\frac{5}{z}\]Ok good.
Hmmm, notice our z is in the denominator, this makes things a little trickier.
Any ideas? :o

- anonymous

Uhhh.... divide five by both sides?...

- anonymous

5/cos(30)= z

- zepdrix

Ya let's try that:\[\large\rm \frac{1}{5}\cdot\cos(30)=\frac{\cancel5}{z}\cdot\frac{1}{\cancel5}\]The z is still stuck in the denominator though!
Ok you had the good sense to flip it after that though?\[\large\rm \frac{\cos(30)}{5}=\frac{1}{z}\qquad\to\qquad \frac{5}{\cos(30)}=z\]

- zepdrix

Or maybe you got lucky :) I'm not sure which lol

- anonymous

XD im pretty sure i got lucky... i didn't even realize i skipped a whole piece.

- zepdrix

When your variable is stuck in the denominator,
these are the steps I would recommend:\[\large\rm \cos(30)=\frac{5}{z}\]Multiply both sides by z,\[\large\rm z\cos(30)=5\]Divide both sides by cos(30),\[\large\rm z=\frac{5}{\cos(30)}\]

- zepdrix

So when your variable is stuck in the bottom,
the process is a little bit different.
Just a bit trickier.

- anonymous

Indeed. When I plugged the equation in, I got 3.8

- zepdrix

Ok, again let's check our work.
The hypotenuse should be the longest side.
But our adjacent side is 5.
Uh oh! 3.8 < 5

- anonymous

My second time i got 5.8 when i did Cos(30), got my answer, then divided 5 by my answer (5/ ans is what im trying to say...)

- zepdrix

Do you have a calculator that uses the "Ans" thing?

- anonymous

Yes i do.

- zepdrix

Ya I really like that feature.
You can do the problem all at once, \(\large\rm 5\div\cos(30)\)
Or in parts as you described,
\(\large\rm \cos(30)\quad \boxed{=}\)
\(\large\rm 5\div Ans \quad \boxed{=}\)
Whichever way makes more sense to you :)

- anonymous

They both do. But sometimes double checking prevents me from making a mistake, gladly.

- zepdrix

I need a math break I think :) lol
Too much maf.
Here is one more you can work on though.

- zepdrix

|dw:1444607625616:dw|

- zepdrix

So if you get bored and wanna try another one, try to solve for x in that triangle :)

- zepdrix

I'mma go make some foods a sec >.<

- anonymous

Alright. I'll attempt to figure this one out. and you enjoy your food. I need a math break after this too XD I have alot more homework... lol

- anonymous

Tan(41)=7/x
7/tan(41)=x
7/tan(41)= 8.1
That's what I got.

- anonymous

Lol then i got 10.7 for the hypotenuse using the same opposite.

- zepdrix

7/tan(41)=8.05, so ya 8.1 sounds good.
10.7? 0_o weird

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