BAM!!! Mr. Tarrou We are going to start taking a look at some right triangle trigonometry.
Now in Geometry this is all new but in PreCalc or Trig which is what I am teaching this section
is actually review. We are going back to the SohCahToa...or SOHCAHTOA!!!! That is the sine
of theta is equal to opposite over hypotenuse, cosine of theta is adjacent over hypotenuse,
and tangent of theta is opposite over adjacent. Now if you have been keeping up my videos,
you might have noticed that a lot of the calculations that we have been doing with these trig functions
lately have been y over r, x over r, and y over x. We came up with these labels when,
and they are very useful, when we are dealing with angles in standard position. We are not
going to be dealing with angles in standard position with this video. We are just doing
right triangle trigonometry, drawing triangles just floating around in space, and with one
of their vertexes on the origin... on the x y axis. These labels are great for standard
position angles but not when you are dealing with right triangles placed done somewhere
in your textbook or on your paper. Let's first start with this question. We have a right
triangle and don't count on these being drawn to scale. We are going to mark off this angle
as theta and outside here a five and thirty. And my question to you is, can we set up the
six trig functions. Part of the reason this is in your book now is to reinforce the idea
that for a lot of trig all we are doing is studying the right triangle. So the six trig
functions are based off the ratios that you get out of the sides of a right triangle and
that means that you need all three of those sides. Once you get all three sides you can
set up any of these three trig functions you want or their reciprocals. We are going to
find six so clearly I did not make my list long enough. But we do need all six sides.
We are going to do that by using the Pythagorean Theorem and that is going to be...what ever
you want to call this. A squared plus b squared equals c squared. That is a squared plus twenty-five
equals 169. We are going to subtract both sides by 25 and we get a squared is equal
to 144. Of course we are going to get rid of the power of two by square rooting both
sides giving us the answer of twelve. If you know your pythagorean triples you might have
already known that this side that I left blank was going to come out to be a length of 12
anyway. So Pythagorean Theorem is going to be what you use to find a missing side of
a right triangle when you know two sides and only missing one side. You are looking for
a third side. Now I did say that we want to find all six trig functions so let me squeeze
this in here. I have plenty of room. What is the reciprocal of sine? It is the cosecant.
The cosecant of theta is going to be the hypotenuse over opposite. The reciprocal of cosine is
the secant. The secant of theta is hypotenuse over the adjacent side. Tangent of theta is
equal to equal to....excuse me. The reciprocal of tangent is cotangent and that is adjacent
over opposite. If you have happened to found this video but you are only in Geometry or
Algebra 2 and just studying the basic three trig functions your textbooks I am not sure
will be using these greek letters, you will be using the capital letters that are on the
corners or vertices of your triangle. So you might see like the cosine, excuse me the sine
of A, the cosine of A, and the tangent of A. I will be using those labels as well probably
in a couple of my examples. I just have a generic triangle with the sides given and
the theta, the greek letter, as the number of degrees or radians in this angle. Now we
are not going to find out what theta is, I just want to set up the six trig functions.
So, let's make sure we know how to label this triangle. If we are using theta, then we need
to name the sides based on this angle that we are going to reference in our six trig
functions in terms of opposite, hypotenuse, and adjacent. The hypotenuse is always opposite
the right angle so 13 is the hypotenuse. Opposite the angle we are going to be working with
is the opposite side of 5. The leg that is touching the angle that using to set up your
trig functions is called the adjacent side. If I wanted to set up trig functions based
off of this angle, then I would have to move these names because this side is not adjacent
to this angle over here...if we were but I am not going to so maybe I should not even
have mentioned that. The sine of theta is opposite over hypotenuse so it is going to
be 5/13. The cosine of theta is adjacent over hypotenuse so we have 12 over 13. The tangent
of theta is opposite over adjacent. Theta here...opposite over adjacent so we have five-twelves.
Now we are going to set up the reciprocal functions which means we are simply going
to flip these three and keep them paired up to help you remember them. The reciprocal
of sine is cosecant and that is thirteen over five. The reciprocal of cosine is secant is
thirteen over twelve. The reciprocal of tangent is the cotangent and that is going to be 12
over 5. Double check...Alright BAM!!! Moving on. Finding the six trig functions from the
sides of a right triangle. Next example! Well the next example is going to be finding the
sides...the missing sides of a right triangle when we only know one side and one angle.
Again probably not drawn to scale but here is a right triangle and we have an angle of
27 degrees, and we have a length of 200, and we are going to look for the side opposite
of 27 degrees which is "a" and the hypotenuse which we have marked as b. Now, if I am finding
the side marked lower case a. By the way I am using lower case letters just to remind
you if you are in Geometry or my PreCalculus kids getting used to this notation, lower
case letters denote the sides of a triangle and the upper case letters are the opposite
angles. So if this is side, this is side A. If this is side "b", then this is angle B.
What is left? C. This is probably not very good labels for Pythagorean Theorem because
you are probably going to be thinking that c is the hypotenuse... but that is ok, that
is where I put the letters so whatever. I will stick with this for now. How are we going
to find... Let's see, 27 degrees. What trig function are we going to use with 27 degrees
to find out what "a" is? When you decide what trig function you like to use, Sine...Cosine...or
Tangent, I like to emphasis the idea of just "What do you have and what do you want?"...
What do you have and what do you want. Well from 27 we want the opposite side and we have
the adjacent side. So we want, we have. We want the opposite side and we have the adjacent
side. Opposite adjacent, opposite over adjacent, that is the tangent function. The tangent
of 27 degrees is equal to the opposite side which is "a" over the adjacent side of 200.
I want to solve for "a", I want to find out how long that side is. I am going to multiply
both sides by 200 to undo that division and get 200 times the tangent of 27 equals a.
Just simply type this into your calculator and, if your calculator is in degree mode
which it will need to be because that is how we labelled this diagram, you are going to
get answer of 101.9 for side "a". Every time you touch sine, cosine, or tangent on your
calculator, and the TI-NSPIRE will let you do all six trig functions, so if you use one
of those trig functions on your calculator make sure that you are in the right mode.
You might get this all set up correctly and get a wrong answer because you are in radian
mode when this question is in degrees. Ok, next example. Now we know that this side is
101.9 and I want to show you the work for finding the hypotenuse. Now, how could we
find the hypotenuse? We could do Pythagorean Theorem, a squared plus b squared equals c
squared, of course the way this is labelled annoying that would be wrong. Let's try the
two legs squared and added equals the hypotenuse squared. But what if this answer is wrong.
We just calculated this ourselves. It is not given information so if this answer has a
mistake in it, and I use it to find another side I am just going to carry that mistake
through the rest of my work. It is always advisable, if possible, to only use the information
given to you in the problem. That is not always possible, but it is advisable. How can I use
the 27 degrees and the 200 units given to find the hypotenuse of b? Well again I am
not going to use the 101.9. From the 27 degrees what do we have and what do we want? Well
if I want to avoid using the 101.9 I have the adjacent and I want the hypotenuse. What
uses adjacent and the hypotenuse? Sine is opposite over hypotenuse, cosine is adjacent
over hypotenuse...so we are going to use the cosine function. The cosine of A which is
27 is equal to the adjacent side of 200 divided by the hypotenuse of b. Now this will confuse
some of my students because the variable is in the denominator. Once I show them once
it is usually ok, but at first they are like "That is kind of weird" You cannot solve for
a variable when it is in the denominator so we need to get it out of that denominator.
Well denominators or fraction bars mean division, the inverse of dividing is multiplication.
So multiply both sides by b and get b times the cosine of 27 degrees equals 200. Now we
have b times a number is equals something else. If I said 2x equals 10, you would divide
both sides by two. So to undo that multiplication we are going to divide both sides by the cosine
of 27. That is going to cancel out and b equals 200 divided by the cosine of 27 and if you
get that typed into your calculator correctly and if I typed it into my calculator correctly
it is approximately 224.5 units. Should we be happy with that answer. Yes we should.
The side that are opposite the largest angles should be the largest side. This is 27, this
angle here is 90 minus 20 is 70 minus 7 is 63. Twenty-seven is the smallest angle and
it is opposite the smallest side. This angle, whatever I just said it was, sixty-three degrees
and it is opposite 200 units, and this largest angle of 90 degrees is opposite the largest
side of 224.5 units. Now, in the next little video I am going to find out how to find an
angle measure in a triangle. I could run through this quickly but I want to make sure you all
know how to use your calculator. So we have a 3,4,5 right triangle and we are going to
find out what theta is equal to. If you already know how to do that, don't watch the next
video! It is going to be short. If you are not sure how to find an acute angle in a right
triangle, the next video will be about five minutes. I will be right back!!!