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And it is indeed a right triangle.
So I'll give you a few seconds to think about that.
Well to answer this you just have to remember the definitions of the trig functions.
To help us there, we'll use the mnemonic soh cah toa.
soh cah toa
And the part of soh cah toa that relates to cosign is the cah part.
This defines sine, that's why we have the 's'; this defines cosine, that's why it starts with a 'c'; this defines tangent, that's why it starts with a 't'.
So if you look at cah, it says that cosine (let me use that same color)...
it says that cosine of an angle is equal to the adjacent side over the hypotenuse.
So in our example here, what is the adjacent side?
Well, if we look at it, it's the side that's next to it and not the hypotenuse.
This side is next to it and it's not the hypotenuse.
This side up here is next to our angle, but it is the hypotenuse; it's the one that's opposite the right angle.
So this is the hypotenuse up here.
This is... if we're looking at angle theta, this is the adjacent side.
And, while we're at it, if you want to think of the oppisite side (we don't have to deal with it for cosine), but it never hurts to label it right now. That is the opposite.
And this is relative to angle theta.
So, with that out of the way, we say that cosine of theta is equal to adjacent over the hypotenuse.
Adjacent has length 4. What is the hypotenuse?
Well we know what side is the hypotenuse, but they haven't given us the length yet.
But we can figure it out using the Pythagorean Theorem.
We have 2 sides of a right triangle, we can always figure out the third side.
We know that the sum of the squares of the 2 shorter sides will be equal to the square of the hypotenuse.
So, we have 4 squared, plus 7 squared is going to be equal to, I'll just call it h squared or hypotenuse squared.
If 4 squred is 16, and 7 squared is 49 is going to be equal to h squared.
And, let's see, 16+50 would be 66, so 16+49 is 65.
So this side over here is 65. H squred is equal to 65.
Or we can say that h is equal to the square root of 65.
And it doesn't look like there's any perfect squares in here - 65 is 13 times 5, neither of those is perfect squares, so this is about as simplified as we can get this radical.
So the hypotenous is equal to the square root of 65.
So in this case cosine of theta is equal to the adjacent side, which has length 4, over the hypotenous which has the length square root of 65.
Now, let's do the same thing with the sine. What is the sine of theta going to be?
I'm going to give you a few seconds to think about it.
Well, soh tells us that sine is equal to opposite over hypotenuse.
In this case, relative to angle theta, the opposite side has length 7.
And what is the hypotenuse, or what is the length of the hypotenuse? Well, we just figured it out.
It is the square root of 65.