BAM!!! Mr. Tarrou I would like to take this opportunity to attempt to help you understand
why cofunctions are equal to each other and actually what are cofunctions. So we are going
to start off with an example here. I want to figure out what is equal to, the cofunction...and
I am not really going to tell you what it is right of the bat, but I want to find another
trig function that has the exact same answer as the sine of 12 degrees. And I want to do
this without the aid of a calculator. I just want you to by the end of this video go, well
the sine of 12 is equal to blah..blah...blah, blah blah blah, only actually have real words
in that sentence. So what are... What are we studying? By the way we are going to be
looking in this video, not angle in standard position but just regular triangles just floating
around in space. Actually do a lot of review of, or a little review of what you learned
in Geometry. That was from SOHCAHTOA!!! Sine is opposite over hypotenuse, sine is equal
to adjacent over hypotenuse, and tangent is equal to opposite over adjacent. Again what
are mainly studying in trigonometry? Doo...dooo... Yes, you are absolutely, you are correct.
We are studying right triangles and their rotations around the inside of a circle mostly.
But we are certainly for the vast majority of the studies of trig, talking about right
triangles. So when we see the sine of 12 degrees, that must mean that we are talking about a
right triangle. Now what do you put into a trig function? You put in an angle measure,
whether it is in degree like this or radians. And what do you get out of a trig function?
Well you get out the sides of a triangle. Now again, I am not going to do y over r because
these are not standard position angles. I am just going to draw a triangle just on the
board somewhere so I am going to go back the SOHCAHTOA. And so the sine of an angle measure
is equal to opposite over hypotenuse. So we must have a triangle that has a right angle
of course because it is a right triangle. It must have an angle of 12 degrees somewhere.
Now I don't care if you draw it like this or the 12 degrees is down here. It makes absolutely
no difference. So I want again to find another trig function that is equal to the sine of
12 without the aid of a calculator and just going though a bunch of calculations. Well,
to help explain exactly cofunctions are always equal I want to take this example, I am not
going to do a proof but a good explanation, and I just want to label the corners of this
triangle and call it ABC. Ok, now what I would like to do is label the sides. Now this is
side BC but how can I label this side in terms of only one letter. You remember that from
Geometry? You use capital letters for angles. Now we us the theta, the Greek symbols for
the rotation or the angles in Trig and PreCalc, but in Geometry we use capital letters...the
vertices. Lower case letters for the sides. A matching side is opposite the given angle,
so this is angle A and this is going to be side 'a'. If this is capital B, angle B, then
this is going to be labelled as little 'b'. Then C, well then this hypotenuse is going
to be little c:) Ok, so yes. Sine is opposite over hypotenuse, but now that I have this
triangle labelled let's loose the opposite over hypotenuse and actually fill in the sides
that are appropriate for this triangle. Now sine, now I just erased it but opposite over
hypotenuse, so from A... the 12 degree angle... the opposite side is 'a' and the hypotenuse
is 'c'. So the sine of 12 is equal to 'a' over 'c'. You would not write this at all
in your actual answer. I am just explaining the concept to you. Now we are going to get
to it. What trig function equals the sine of 12? Well, if we are allowed to use this
triangle. And be aware that we never set up trig functions off of... at least in this
kind of setting trying to learn trigonometry... We never set up trig functions based on the
90 degree angle. Indeed if you looked at my unit circle videos, we can only set up reference
triangles for those angles that are in quadrants 1, 2, 3, and 4. The 30, 45, and 60 degree
angles, The 120, 135, and the 150 angles and so on. The 90, 180, 270, and the 360 or 0,
those are quadrantal angles. It is not possible to set up reference triangles for those angles.
So we are not going to be setting up trig functions off of the 90 degree angle. But
we have 12 degrees and we have 78 degrees over here for angle B. Can you think of a
sine, cosine, or tangent function using either one of these acute angles that is going to
have the same ratio...the same answer as the sine of 12. Well, how about cosine? The cosine
of... Well I cannot do the cosine of 12 because that would be b over c. But if I do the cosine
of 78 degrees, you know... Cosine is adjacent over hypotenuse, whereas sine is opposite
over hypotenuse. So those are not going to be the same ratio, but if I am changing to
the other acute angle inside a right triangle... from 78. The cosine of 78 is adjacent over
hypotenuse, so it is still going to be a over c. Do you see what is happing there. From
12 it was opposite over hypotenuse and from the 78 degrees a/c is the adjacent over the
hypotenuse. So the sine of 12 equal the cosine of 78. It is no coincidence that 12 plus 78
is equal to 90. It is no coincidence that these two angles are complimentary. Because
how many degrees are inside any triangle? 180. Well in a right triangle, 90 is taken
up with the right angle inside that right triangle. That only leaves you 90 degrees
left for the other two acute angles to share. So those other two acute angles are always
going to be complimentary. So what I just showed you, not a proof but just an example...
close to a proof, is that the sine of 12 equals the cosine of 78. Or indeed, the sine of some
given angle is equal to the cosine of the compliment of that angle or 90 minus theta.
This is your first pair of cofunctions. Now I am going to go a little bit faster and just
give you the formulas. Because the work is very very simple for this concept. I just
wanted to take a few minutes and explain why it is true. Another formula is the cosecant,
that is the reciprocal of sine, is equal to the secant, the reciprocal of cosine, of 90
minus theta. These are also... These are the second pair of cofunctions. Then finally,
the tangent of theta is equal to the cotangent of 90 minus theta. This is your third pair
of cofunctions. These will work in reverse as well. So your textbook will either give
you three little versions of this, or maybe six to say something like cosine of theta
equal the sine of 90 minus theta. I like to summarize these little tiny formulas into
a single statement. That is cofunctions are equal when their angles are complimentary. Alright!
So there you go. That means that if I say, HEY what is equal to the tangent of 30 degrees?
Or what is the cofunction to the tangent of 30 degrees? That is just the cotangent. What
is 90 minus 30? 60 degrees. What is equal to the cosecant of pi/3. I am going to cheat
and use these same degrees only in radian version. The cosecant of pi/3, which is 60
degrees, is going to be equal to the cofunction to cosecant is secant. So the cosecant of
pi/3 is going to be the secant of pi/2...that is 90 degrees... minus pi/3. If you know the
unit circle you might already know the answer is pi/6 because 90 minus 60 is 30. But you
can find common denominators. And I just mis-wrote that a little bit. You get the secant of 3pi
minus 2pi over 6. That comes out to as we already knew, the secant of pi/6. So, these
should be the very very easy questions of your next quiz or test. The sine of 10 equal
the cosine of 80. The tangent of 20 equals the cotangent of 70. I am starting to forget
what I said. Cofunctions are equal when their angles are complimentary. So that is the rule.
I hope you understand why they are equal now. BAM!!! I am Mr. Tarrou. Go Do Your Homework!!!