Pre-Calculus: Trig to find a Right Triangle angle

This lesson has not been reviewed.

Please purchase the lesson to review.

The Trigonometric Functions
Right Angle Trigonometry
Evaluating Trigonometric Functions for an Angle in a Right Triangle Page [1 of 2]
Let?s take a look at some of these trig functions in action with a couple more examples of different types. First of all,
let?s suppose that I just give you this triangle. It?s a right triangle, that?s a good start. This side is two, this side is four,
and this here is the angle ?. And let?s find all the trig functions for this angle. So, I want to find sine, cosine, tangent;
let?s even find cosecant, secant and cotangent for this angle.
Well, it looks like a tangent won?t be a big deal, because that?s just opposite over adjacent, so that?s going to be pretty
good, but what about sine and cosine? I need to know the length of the hypotenuse. I don?t know what that is. That
wasn?t given to us. Well, can we find it? Sure, we?ll just use the Pythagorean Theorem. Using the Pythagorean
Theorem, what I have is the following. I?m just redrawing my picture. This is two, this is four, I don?t know what this is,
I?ll just call it ?h? for hypotenuse. I know that this squared plus that squared has to equal the hypotenuse squared. So,
the hypotenuse squared has to equal the 42 , which is 16, plus 22 , which is 4. Therefore, h 2 has to equal 20. So, I
could take plus or minus the square root. Here, I can rid of the minus square root because, remember, this
hypotenuse is a length. Length is always positive. So, here I can dismiss the negative square root as an extraneous
root and I would see that h = 20 . And that?s fine by itself, or you could realize that the 20 is just 4 multiplied by
5, and by laws of exponents, I could just take the square root of each of those terms, the 4 x 5 , and the 4 is 2,
so this is just 2 5 . So, knowing two sides of the right triangle, I can actually find the third side, In fact, that would
have to be 2 5 .
Great. Well, now I can just rip through all of the trig functions. So sine of ?, that?s opposite four over hypotenuse,
2 5 , so I can cancel a little bit and see just 2 on top over the 5 . What about cosine ?? Well, that?s adjacent over
hypotenuse, which is just 1 over the 5 , after you cancel. Tangent would be opposite over adjacent,
4
2
, which is
just 2, and what about the reciprocal ones? Let?s do cosecant. Cosecant, remember, is just he reciprocal of sine of ?,
so I just flip this and I see
5
2
; secant is just the flip of cosine, so that?s just the
5
1
; and cotangent, the only one
that?s sort of easy to remember, is the reciprocal of 2, which would be
1
2
. So, in fact, there?s all the trig functions, just
knowing two sides, I was able to find all the trig functions of this angle.
Let?s try a different type of question. Suppose that I tell you that ? is some acute angle. So, it?s between zero and
ninety. And that the sine ? equals
12
13
. I want you to find all the other trig functions for that same angle. Well, how in
the world could we possibly find all the other trig functions for that angle, that we know is acute, just knowing it?s sine?
It seems impossible. Until we start thinking about, since this thing is acute, we could think about it as being one of the
angles in a right triangle. So, let?s just draw a picture. If we draw a picture, and pick an angle to be the angle ?, let?s
say this one right here. It doesn?t make a difference which one. You can?t pick it to be the 90° one, but you pick it to
be either of these two. Suppose I say it?s that one. Well, if the sine of ? equals
12
13
, that means I could take my
triangle so that the proportions of ? now what?s sine? It?s opposite over hypotenuse ? would be in that ratio. So, I
could say this thing has length 12, and that forces this to have length 13. Hey, well this is now just the thing that I just
did. If I know two sides, I can find the third side of a right triangle, using the Pythagorean Theorem.
So, how would the Pythagorean Theorem look here? Now I want to find this length, that?s a leg, let me call it ?l? for
leg. It?s not the hypotenuse and so what do I know? I know that this squared plus that squared should equal that
squared. So, that means that l 2 + 12 2 , which is 144, should equal 13 2 , which is 169. So that means that l 2 should
equal, if you bring this over, 25. So I take square roots, but I have to take plus or minus square roots, I could always
get two answers, but in this case, I can just take the positive one because this is a length and the square root of 25 is
just 5. So, in fact, this length must be 5. Well, great. Well, now I can, in fact, read off all of the other trig functions? I
The Trigonometric Functions
Right Angle Trigonometry
Evaluating Trigonometric Functions for an Angle in a Right Triangle Page [2 of 2]
already know sine, opposite over hypotenuse. What?s cosine? Cosine is adjacent over hypotenuse, so that would be
5
13
. What?s tangent? That would be opposite over adjacent, so
12
5
. What would be cosecant? Well, that?s just the
reciprocal of sine, so that would be
13
12
. Secant would just be the flip of cosine, so
13
5
, and, let?s see, cotangent
would be flip of tangent, which would be
5
12
. So, that?s pretty cool. Just knowing the value of an acute angle, I can
find the sine of the value of an acute angle. I can find all of the other trig functions because I can actually reconstruct
the triangle that has that as it?s sine and then read off all the other ratios. Pretty cool.
Okay, up next, we?re going to take a look at more connections, sort of, between triangles and angles and actually
seeing how we can take the angle of something, the trig function of something and correspond it with it?s triangle and
start to get information back and forth about each of those things. We?re just sort of getting our feet wet here on this
trig stuff, but the idea is to build up some intuition and see connections with the geometry and the trigonometry here
and then just start to also find out the values, the trig functions, of certain special angles. We?ll use those later. So,
anyway, a lot is going on here, and it?s all trig. Fun. I?ll see you soon.