Bo: Hey, guys.
Billy: Hey, Bo!
Bobby: Hi, Bo.
♫ (lyrics) Flipping Physics ♫
Mr. P: Ladies and gentlepeople.
The bell has rung, therefore class has begun.
Therefore, you should be seated in your seat,
ready and excited to review SOH CAH TOA,
and the Pythagorean Theorem.
Billy: Oh, boy!
Bo: Shouldn't we already know those.
Mr. P: Bo, that is certainly true.
Before taking this algebra-based physics class,
you should have already learned this,
however, it is such an important piece of physics
that I feel it necessary to review.
Bo: Okay. That makes sense.
Mr. P: Let's start with a triangle.
Billy, what kind of triangle is this?
Billy: A right triangle.
Mr. P: Bobby, what is a right triangle?
Bobby: A triangle with a right angle?
Mr. P: Does that mean that there are triangles
with wrong angles in them?
(Bo laughs)
Bobby: I suppose not.
A right angle is a 90-degree angle,
so a right triangle is a triangle that has
a 90-degree angle in it.
Mr. P: Good.
This symbol is the symbol for a right angle,
or a 90-degree angle, and that makes this
a right triangle, or a triangle that has a 90-degree angle in it.
Let's identify some parts of this triangle.
Let's call this side X, which has a distance of 4.7 meters,
this side of the triangle, Y,
this angle of the triangle is going to be theta-1,
and that has a measurement of 33 degrees.
This is symbol theta, it's a Greek letter.
It's a Greek letter theta,
and it's commonly used for angles in physics.
Let's call this side H for hypotenuse,
or the side opposite the right angle,
and let's identify this angle as theta-2.
Our goal is to determine the values for Y,
the hypotenuse, and theta-2.
Billy, how would you like to begin?
Billy: The interior angles of any triangle
add up to 180 degrees,
so we just need to ...
Mr. P: Hold up.
Let me interrupt you for a minute, Billy.
You are welcome to do it that way.
In fact, it would be easier to do it that way.
However, right now, I'm trying to review SOH CAH TOA,
and the Pythagorean Theorem,
and what you are doing actually obviates that.
On a quiz or test, you should certainly solve it that way.
However, not right now.
Let's actually start by just defining SOH.
Bo, what does 'SOH' mean?
Bo: SOH means sine opposite over hypotenuse.
Mr. P: Like that?
Bo: No, sine equals opposite over hypotenuse.
Mr. P: Right.
How many of you think SOH means
sine equals opposite over hypotenuse.
Great.
This is one of the reasons we have to review this concept
because you are all wrong.
Like most students, you are skipping steps,
just plain old dropping out useful information.
It's not there in the equation.
You're just leaving it out.
Bobby: But that's how I've always done it.
Bo: Yeah.
Mr. P: You have left out a very important
part of the equation.
SOH means sine theta equals opposite over hypotenuse.
Billy: Pish tosh, it's the same thing!
Bo: Pish tosh.
Mr. P: It is absolutely not the same thing.
I've seen it time and time again
that students who leave out the theta
make mistakes when solving problems like this one.
I don't know why you all eschew theta,
however, you just can't,
so please remember to write out the whole equation.
Bobby, what does 'CAH' mean?
Bobby: Adjacent over hypotenuse.
Cosine theta equals adjacent over hypotenuse.
(Billy and Bo laugh)
Mr. P: And Billy, 'TOA'?
Billy: Tangent theta equals opposite over adjacent.
Mr. P: Great.
Bo, what does opposite mean?
Bo: Huh?
Mr. P: What does opposite mean?
Bo: It means opposite.
Mr. P: Opposite what?
Bo: Oppos- ...
Oh, opposite to the theta, to the angle.
Mr. P: Right, opposite to the angle, or your theta.
Notice that the opposite and adjacent
refer to the sides that are opposite and adjacent
to the angle you are referring to.
Be very careful to always check to make sure
you're using the correct sides as opposites, etcetera,
because it does depend on which angle you are using.
The hypotenuse is always opposite the right angle,
so, again, this side here is opposite of this angle,
and this side is adjacent to this angle.
However, if we're talking about this angle or this theta,
this side is adjacent,
and this side would be opposite that theta,
so please be very careful.
Actually, I think we were doing a problem.
Let's get back to that original problem.
Bo, you were working on this.
How would you like to begin?
Bo: (sighs) Well, you know cosine of theta
equals 4.7 over the hypotenuse.
Mr. P: Another basic tenet of how I teach
is that you have to show your work.
This means that you must start with an equation
and variables.
You can't skip steps.
You can't start with numbers.
You must start with variables.
Please try again.
Bo: Okay, we know cosine theta
equals adjacent over hypotenuse.
Mr. P: Yes.
Bo: And we can use theta-1,
that means that our equation would be cosine of theta-1
equals X over the hypotenuse,
now we know theta-1,
we know the values for theta-1 and X,
so can we plug in our values now?
Mr. P: Yes, now we have written out the equation
and substituted in our variables,
we can substitute in our numbers.
Bo: That means that cosine of 33 degrees
equals 4.7 over the hypotenuse.
Mr. P: Billy, how do we solve this equation for the hypotenuse?
Billy: Multiply both sides by the hypotenuse
and divide both sides by the cosine of 33 degrees.
Mr. P: Absolutely.
Notice how the hypotenuse cancels out,
and then cosine of 33 cancels out,
and we end up with the hypotenuse
equals 4.7 divided by the cosine of 33.
Bobby, what do we get for an answer?
Bobby: Negative 354?
Mr. P: Be careful,
you need to make sure your calculator is in degree mode.
Your calculator is in radian mode,
so please be careful of that
because our angles are in degrees,
we need our calculators in degree mode.
Bobby: Oh, dangit!
It's 5.6041,
and because the least number of significant figures
from our givens was 2,
we need to round to 2 sig figs, or 6.5, no 5.6 meters.
Mr. P: Very nice.
Now we need to find either Y or theta-2.
Bobby?
Bobby: Can't we just use the Pythagorean Theorem now?
Mr. P: Yes, how do we know we can use
the Pythagorean Theorem?
Bobby: Because it's a right triangle.
Mr. P: Yes.
A² plus B² equals C².
The Pythagorean Theorem,
and please notice that the C is always opposite
the right angle,
because it represents the hypotenuse.
Bobby: Okay, A² plus B² equals C².
So X² plus Y² equals the hypotenuse²,
and to solve for Y, we subtract X² from both sides
and then take the square root of the whole equation.
That gives us Y equals the square root
of the quantity H² minus X².
Plugging in our numbers,
we get Y equals the square root of the quantity
5.6² minus 4.7².
That works out to be 3.0447,
which rounds to 3.0 meters with two sig figs.
Mr. P: We made a mistake.
Bobby: Dangit, again!
I used the rounded answer to solve the problem.
We need to use the unrounded number.
We need to use 5.6041 instead of 5.6.
that works out to be 3.0522, which rounds to 3.1 meters
with two sig figs and not 3.0.
Mr. P: Yes. Please be careful not to use rounded numbers
when solving a problem.
This is why I always write out my unrounded answer first
and then I round to get my answer.
That way, I have the unrounded answer if I need it.
Now, we have found Y and the hypotenuse.
Billy, could you please help us find theta-2?
Billy: Tangent theta equals oppostie over adjacent,
so tangent of theta-2 equals 4.7 over-
Mr. P: Ah!
Billy: 3.0522 ...
Mr. P: Ah!
Ahh!
Steps.
You need to make sure
that you take every single step.
If you do not take every step,
if you start skipping steps,
you are going to fall down the stairs and get hurt.
There's going to be blood on the stairs
and I will have to call the janitor
so that he can get his PPEs to take care of your BBPs.
PPEs to take care of your BBPs.
Personal protective equipment to take care of your
blood borne pathogens.
Don't skip steps.
You are going to get angry with me.
I know this.
You're going to be like,
"But there's a handrail, Mr. P!
"If I hold onto the handrail, can't I just skip one teensy step?"
No.
You started with an equation, which is great.
However, you need to now plug in the variables,
not the numbers.
Class, if you just plug in the numbers,
what would you be doing?
Billy: Skipping a step.
Bobby: Falling down the stairs?
Bo: Using the handrail?
Mr. P: Write out the equation.
Substitute in the appropriate variables,
then substitute in the numbers.
Billy, we're back on you.
Billy: All right, we already have the equation written down,
so now I guess we substitue in variables.
Tangent theta-2 equals X over Y.
Mr. P: Great!
Notice how important that subscript of 2 on the theta is.
If the subscript were a 1 instead,
the equation would have Y over X
instead of X over Y.
Billy: Now we can substitute in numbers
and get 4.7 over 3.0522.
Mr. P: Bobby, how do we solve for theta-2 in this equation?
Bobby: I remember.
Take the inverse tangent of the whole equation,
both sides of the equation.
Mr. P: Yes, Bobby.
Let's walk through what that looks like on the board.
On the left-hand side of the equation,
you get the inverse tangent of tangent of theta-2.
Bo, what does the inverse tangent
of the tangent of theta-2 actually work out to be?
Bo: Oh yeah, the inverse tangent of
the tangent of theta-2 just works out to be theta-2.
Mr. P: Right, theta-2, and on the right-hand side,
we get the inverse tangent of 4.7 over 3.0522.
Bo, what then does the answer work out to be?
Bo: Wow.
It worked out to be exactly 57 degrees.
So with two significant digits,
theta-2 equals 57 degrees.
Billy: Mr. P, why didn't I get exactly 57 degrees?
Mr. P: Bo must have used the answer button
on his calculator rather than retyping the number in
with fewer significant digits.
I always recommend using the answer button
because you get a more precise answer.
Okay, please remember that you must have a right triangle
in order to use SOH CAH TOA
and the Pythagorean Theorem.
Thank you very much for learning with me today,
I enjoyed learning with you.
Voiceover: Lecture notes
are available at FlippingPhysics.com.
Please enjoy lecture notes responsibly.