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# Finding trig functions of special angles example

Video transcript

Voiceover:The first thing
I want to do in this video is given the information
given about this triangle. Figuring out the unknowns, figuring out the length here this value a, figuring out this length in orange, b, figuring out the measure of this angle which we'll call alpha in
figuring out in radians and figuring out the
measure of this angle, right over here in blue
that we could call beta. I encourage you to pause the video now and given the information that we have try to figure out these unknowns. Assuming you've given it a try, so let's see what we can determine. We know two of the side lengths and they're the same, they
both have side length two and they've given us one of
the angles right over here, this yellow angle has a measure
of pi over three radians. One thing that jumps out of me is that this is an isosceles triangle. An isosceles triangle has two sides that have the same length so this side is congruent to
that side right over there. That means that the base angles, the corresponding base angles are also going to be congruent and that's just because this is a property of isosceles triangles. These two are congruent so the two base angles
are going to be congruent. That means that beta is also equal to pi over three radians. This is also beta is also equal to pi over three radians. Now let's think about ... Now what else we can
attempt to figure out. Well if we know beta is pi over three and we know this angle in yellow is pi over three radians, we can figure out what
this angle over here in white is and I don't know, let's call that angle, the measure of that angle let's call it x. If we're thinking in terms of radians x plus, this angle plus
pi over three radians, plus pi over three radians, this is the sum of the
interior pi over three. This is the sum of the interior angles of this larger triangle right over here, they're going to sum up to pi radians. If we're thinking in degrees
it would be 180 degrees so those are going to
sum up to pi radians. What's x going to be and this one you can
almost look straight at it. You say, this is going to be pi over three but if you want to solve a
little bit more systematically you can subtract these two from both sides and you get x is equal to, I'll rewrite this orange
pi is three pi over three, this is the same thing as
pi, minus pi over three, minus pi over three, which of course three pi minus pi minus pi is going to be pi over three. These angles measure is
also pi over three radians. Now that's interesting
because these shows us that this is actually
an equilateral triangle. All of the sides, when all of the sides have the same angle measure that means … when all the angles have
the same angle measure that means all the sides
are going to be congruent. If this side is two, this side is two then that means that this
entire side right over here, this entire side right
over here is equal to two. Now what else can we
attempt to figure out. Well let's see we know this is pi over three radians. We know that this is pi, we know beta is pi over three, we know that this is … If we're thinking degrees 90 degrees which is the same thing as pi over two so if we know these two
angles of a triangle we can figure out what alpha is. We know that alpha, let me write it here. Alpha plus pi over three, plus this right angle which
in radians is pi over two. We'll do in white. Plus pi over two, that's
this angle right over here, plus pi over two are going to equal … or the sum of the interior
angles of a triangle ... Remember we're looking at
this triangle right over here that's going to be equal to if we thought in degrees it
would be 180 degrees in radians it would be equal to pi. We get alpha, alpha is equal to pi which
is the same thing ... Let's see I'll find a
common denominator of six. We can write that as six pi over six, minus pi over three which
is the same thing as two pi over six. I'm just subtracting this from both sides and I wrote it like that. Minus pi over two which
is the same thing as three pi over six and that's going to be equal to six minus two minus three is one. That's going to be equal to pi over six. Alpha is pie over six so let me write that alpha is
equal to pi over six radians. This over here is pi over six radians. Now what's interesting about that is that shows that alpha is exactly half of this pi over three radian,
white angle over here. That has measure angle at alpha and this one also has alpha, a measure of alpha radians. Pi over six plus pi over
six is pi over three. Now what's interesting about that is that gives us confidence
that this thing over here is going to be bisecting
this side over here that this length is equivalent to this
length right over here. We are bisecting this angle and so when we get to the other side and this is an isosceles triangle, we could actually now prove that these are congruent triangles that you have a side, an angle and a side, side angle side. This side corresponds to that side so these two sides need to be congruent. Now because these two sides are congruent we know this has length b then this has also length b so we know that b plus b is equal to two, or we could just say b is equal to one. Now we could use that
information to solve for a using the Pythagorean Theorem. This is a right triangle
so we could say a squared plus b squared which is one squared is equal to two squared, or we could say, let's say a squared is going to be equal to
two squared minus one so it's going to be equal to three. I just subtracted one from both sides and so we would have a is equal to the square root of three. If you remember your
special right triangles from geometry class your
30, 60, 90 triangles you'll recognize that this is one of them but instead of calling
it a 30, 60, 90 triangle we could call it a pi
over six, pi over three, pi over two radian triangle but it really is in degrees 30, 60, 90 and we see it has a properties
of the 30, 60, 90 triangle where the shorter side or the side opposite the
pi over six radian angle has half the length of the hypotenuse and then the longer non hypotenuse side is square root of three
times the shorter side, the side opposite the 60 degree angle or the pi over three radian angle has the length of square root of three times the shortest side. Just like that we were able to figure out all of the missing information in this. Now given that, what I want to do is figure out what the
sine, cosine, and tangent of alpha actually are and I could have done it for alpha or I could have done it for beta but I'll just do it for alpha and actually I'll leave
it as an exercise to you to do it for beta. Let's figure out so the sine of alpha and I encourage you to pause the video and try to think of what this is. Well the sine of alpha is the same as sine of pi over six, and remember Sohcahtoa definition. Let me do Toa in a different color. Sohcahtoa. Sine is opposite over hypotenuse so for this angle right over here the opposite side is one, so it's equal to one over the hypotenuse, one over two. Sine of pi over six is one half. Now what is cosine of alpha or what is cosine of pi over six? Well cosine is adjacent over hypotenuse so the adjacent side to this angle, that's not the hypotenuse,
it's square root of three and the hypotenuse once again is two. The cosine of pi over six is square root of three over two and then finally for the
tangent of pi over six, we could just take it
to sine over the cosine or we could say it's the
opposite over the adjacent. The opposite is one, the opposite angle to alpha is one and the adjacent is square root of three so it's one over the square root of three and if you don't like irrational
numbers in the denominator we can multiply it by
the square root of three over the square of three to get square root of three over three and we are done. We have figured out a
lot about this triangle.