All right. In this video, I want to talk about trig functions and right triangles.
So the idea is you have a right triangle and...
so there's our 90 degree angle there on the bottom right (not the best right triangle in the world),
but I think you get the idea.
Suppose we call the angle in the bottom left corner theta,
and we'll label the side
opposite theta ... we'll just call that O,
we'll call the long side the hypotenuse, we'll call that H, and we'll call the adjacent side A.
So there's a relationship
involving sine of theta,
cosine of theta, and tangent of theta that you can read off of this triangle.
Sine of theta is going to be defined to be the side opposite over the
hypotenuse. Cosine
is the adjacent over the hypotenuse, and tangent of theta is going to be the opposite over the adjacent.
From that we can get cosecant of theta. Since that's just the flip of sine theta, that'll be hypotenuse over adjacent.
Secant of theta - that's the flip of cosine - that'll be (oops -
hypotenuse over opposite - almost didn't catch that, was looking further)
Secant of theta is the flip of cosine. That one will be hypotenuse over adjacent, and
then cotangent of theta is gonna simply be adjacent over opposite.
Okay, so
let's ... let's do some problems involving this stuff here.
So suppose we want to find ...
so let's find, let's say, tangent of theta, and
let's find
secant of theta
if we're given the following. Suppose we're given that sine of theta equals
3/5 and
that theta is between 0 and Pi over 2.
All right, so
if theta is between 0 and Pi over 2 in terms of the circle, that means it's somewhere in the top right quadrant, and,
if you think about the trig functions, all the trig functions are positive there, so tangent of theta should be a positive number.
Secant of theta should also be a positive number.
But all I do here is just make a little right triangle, so
there's theta. Again, sine of theta is defined to be the opposite over the hypotenuse
So that means the opposite side is 3 the hypotenuse side is 5 and then to figure out the missing side,
the adjacent side,
remember we can use Pythagorean Theorem. And it says
if you take the two shorter sides and square those that has to equal the longer side squared.
So 5 squared is 25
and 3 squared is 9
if we subtract 25 minus 9 we get 16
so A squared equals 16.
If you take the square root, normally you get positive and negative, but obviously the length has to be positive
So the adjacent side is 4, so this is one of our familiar 345 triangles.
So if we want to figure out what tangent of theta is..
Again, tangent is defined to be the opposite over the adjacent, so tangent of theta will be 3/4.
Secant of theta
is defined to be the hypotenuse over the adjacent.
So secant of theta will simply be the hypotenuse, 5, over the adjacent, which is 4.
And
there are your solutions.
Let's do another one similar to this.
Suppose, we're given that cosine of theta
equals, let's say negative 1/3.
And let's suppose that theta is between Pi and 3 Pi over 2, so,
again, Pi and 3 Pi over 2 is somewhere in the bottom left quadrant.
And what I do for these, even though it's a negative,. I don't even, when I make my right triangle,
I don't even really think about that.
Okay, so here's theta. Again, cosine...
We said that that's the adjacent
Cosine's the adjacent over the hypotenuse, so that means the adjacent will be 1, the hypotenuse will be 3.
Again, to figure out the missing side, we'll take 1 squared plus the opposite side squared...
(t looks like a zero.
Oh well, it's supposed to be Opposite.)
...equals 3 squared.
So that means the opposite side squared is going to equal 9 minus 1 which is 8, so the opposite side is
gonna equal simply square root of 8.
So the opposite side (I'll put Op down therefore opposite)...
All right, so, again, now the same thing. If we want to just find, again, tangent of theta and,
also, if we want to figure out secant of theta from this triangle, we do the same thing as before.
So tangent is the opposite over the adjacent.
So square root of 8 over 1, and this is where I think about signs. So
tangent is the sine divided by cosine, which is the x-coordinate divided by the
y-coordinate. Well, both the x-coordinate and the y-coordinate are going to be negative in the bottom left quadrant,
so tangent should be positive.
Secant of theta, again,
we could really just flip the value
we have already. But if we didn't think about it that way, if we did hypotenuse,
which is 3, over adjacent, which is 1, and, again, if you think about secant, that's one over cosine.
Cosine is the x-coordinate.
So on the bottom left quadrant, cosine is negative.
Well, one over a negative is also a negative, so secant of theta would simply equal
negative 3. And, again, those would be your solutions.
Let's do maybe one more.
Suppose we want to figure out sine of 2y.
And we're given that secant of y equals
5/4, and suppose that the angle y is between 0 and Pi over 2.
Okay, so the same thing. We know that our angle is between 0 and Pi over 2, so it's in the top right quadrant.
(Sorry squeeze it back in there.)
So, secant, if we make our little right triangle....
So now my angle, instead of being called theta, is being called y.
Secant, again, we said that that was
the hypotenuse over the adjacent.
So that means the hypotenuse is 5,
the adjacent is 4. We could do Pythagorean Theorem again to find, in this case, that the other side is 3.
Well, to figure out sine of 2y, this is where you have to use your trig identities.
There's a trig identity that says sine of 2y is equivalent to 2 [times]
sine y [times]
cosine y. And now we simply read off our values, sine again being the
opposite over the hypotenuse, cosine being the adjacent over the hypotenuse.
While the opposite over the hypotenuse will be 3/5, the adjacent over the hypotenuse will be
4/5, and
you could think about 2 as simply being 2 over 1.
We just multiply across the top: 2 times 3 is 6 times 4 is 24, and on the bottom
25. And that will be our solution.
Remember sine of any quantity has to be something between positive 1 and negative 1 which this certainly is.
But if we had gotten a value, you know, if this worked out to be
7/3 or something, you would know that you definitely had made a mistake somewhere along the line.
So there's a little bit of a basic trig stuff involving right triangles. If you need to see some other...
some other things involving trig, feel free to take a look at my website. I've got a quite a few trig videos with more coming
along with a ton of Calculus and Algebra videos.
If you have any questions feel free to send me an email, and I'll try to address those as soon as possible.