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We're on problem 66. In the accompanying diagram,
the measure of angle a is 32 degrees. And AC is equal to 10. Which equation could be used
to find x in triangle ABC? So we want x. So let's write our
SOHCAHTOA down. What are they giving us? Well x is the opposite side,
this is equal to the opposite, it's opposite of our
angle in question. And then this 10 side
is the adjacent. So I haven't looked at the
choices yet, but if we say the tangent of 32 degrees should be
equal to the opposite side, which is x, over the adjacent
side, which is 10. And then if we multiply both
sides of this by 10, we get 10 tangent of 32 degrees
is equal to x. Let's hope that's one
of the choices. Yep, sure enough, C. x is equal to 10 tangent
of 32 degrees. Everything in trigonometry
really does boil down to SOHCAHTOA. You'll be amazed. When you actually take
trigonometry as a class, you'll see that they
have all these trig identities and all that. But all of those things can be
just proved from SOHCAHTOA. And these are just definitions
of ratios that have been very useful in the world when
studying triangles. All right, 67. But you can watch, we have a
whole playlist on trigonometry and we talk about all the
ratios and the unit circle and all that. So if you're enjoying these
problems, I encourage you to watch those videos. Or if you don't understand these
problems, I encourage you even more to watch
those videos. All right. The diagram shows an 8 foot
ladder leaning against a wall. The ladder makes a 53 degree
angle with the wall. Which is closest to the distance
up the wall the ladder reaches? So this is what they
care about, this distance right here. So let's see how we
can figure it out. If we write SOHCAHTOA, maybe
we don't have to to write that anymore. So we have this angle and
what have they given us? This deals with the hypotenuse,
which is 8. And it deals with this
adjacent side. So what trig function deals with
the adjacent side and the hypotenuse. Well, I'll write it down. SOHCAHTOA. We want to figure out adjacent,
we have the hypotenuse. What trig function
deals with it? Well, cosine is adjacent
over hypotenuse. So the cosine of 53 degrees is
equal to the adjacent side of this triangle, of this
right triangle. So that's A. And you should only be applying
these, right now, with right triangles. Later on we'll learn, in
trigonometry how trig functions are useful
for any triangle. Because obviously, only
a right triangle has a hypotenuse. So these h's wouldn't be too
meaningful if you're not dealing with a right triangle. Back to the problem. Cosine of 53, it equals the
adjacent side over the hypotenuse, which is 8. I can write x there, x
is the adjacent side. So we multiply both sides of
this by 8, we get 8 times the cosine of 53 degrees
is equal to x. And they give us
a little chart. They tell us that the cosine of
53 degrees is approximately equal to 0.6. So we get 8 times 0.6
is equal to x. 8 times 0.6 is 4.8. So x is equal to 4.8. And that is choice B. Problem 68. Triangle JKL is shown below. Fair enough. Which equation should be used
to find the length of JK? This is JK. All right so we have
this angle here. JK is the opposite side. We want to figure out
the opposite side. We have the hypotenuse. So what trig function deals with
opposite and hypotenuse? Well I could just
write SOH down. I'm don't even have to write
the whole SOHCAHTOA. Because sine is equal to the
opposite side over the hypotenuse. So in this case, sine of 24
degrees is equal to the opposite side, which is side
JK, over the hypotenuse. This is the hypotenuse
right here, 28. The longest side of the right
triangle, or the side that's opposite the 90 degree angle. And so maybe we're done. So sine of 24 degrees is
equal to JK over 28. That's choice A. Problem 69. These are kind of fun,
aren't they? What is the approximate height
in feet of the tree in the figure below? Now you'll see why trigonometry
is useful, because I'm sure you often
wonder how tall is that tree? And it might be hard
to climb it. And now you can use
trigonometry. If you just have a way to
measure the angle between your line of sight and between you
and the top of the tree. You've probably seen those
surveyors, actually they use things like this. And you can figure out the
heights of things far away if you know how far away
you are from it. But anyway, let's work
on this problem. What is the approximate height
in feet of the tree in the figure below? OK, so this is 50. So let's think about it. What do they tell us? This is 90, this is 50. So the approximate
height in feet. OK, so this is the opposite
side of this angle. So let's call this h. So what deals with the
opposite side? Actually I should use h,
let's call that O. O is the height of the
tree because it's opposite the 50 angle. So what deals with the opposite
and the hypotenuse? So once again, SOHCAHTOA. We just have to look
at the SOHA part. Sine is equal to opposite
over hypotenuse. So the sine of 50 degrees is
equal to the opposite side of this triangle, that's the
height of the tree. So I'll say O for the
height of the tree. Over the hypotenuse, the
hypotenuse is 100 feet. Multiply both sides by 100. You get 100 sine of 50 is equal
to this opposite side, which is equal to the
height of the tree. Sine of 50 is 0.76. So 0.76 times 100
is equal to O. That 76 is equal to O, or
the height of the tree. Oh, they said 0.766. So let's put another 6. That becomes 76.6. And that's choice B. All right. Problem 70. These go fast. If a is equal
to 3 roots of 3 in the triangle below. I think you're going to learn a
little bit about 30, 60, 90 triangles now. What is the value of B? OK, maybe they assume that you
already do know something about 30, 60, 90 triangles. But anyway, if a is equal
to 3 root 3, what is B? And I'm debating on what
I should assume that you already know. But let's think about
it a little bit. Let me see how I can do this
without resorting to just the definition of 30, 60,
90 triangles. I'll prove it for you. Although, later you
can memorize. Because when I keep saying 30,
60, 90 triangles, we know that this right here is 60 degrees. Because 30, plus 60, plus 90
is equal to 180 degrees. So that's why we could
we keep calling this a 30, 60, 90 triangle. So how do we figure out
this side, if we just know that side? And we know this
is 30 degrees. So what I'm going to do is,
I'm going to try to redraw this same triangle but I'm
going to flip it over. And this is actually the proof
of figuring out the measures of a 30, 60, 90 triangle. So maybe I'll do it in
a different color. And I have some time
so I'll do it. Let's see. So I'll do it like that. I have to bring another line
down, something like that. And then I'll have to draw
another line more like that. I think you get the idea. And then bring a line
across like that. And actually, I'm going to draw
another line straight down like that. OK, let's see what we
can do with this. Maybe there's an easier
way to do it. But this is just what my brain
is thinking of right now. Let's think about
it a little bit. In this drawing that I've done,
this triangle is just a mirror image of that one. So this right here,
this is also 60. This is 60, this is 60, what's
this angle going to be? Well they're all collectively
supplements of each other. If you go all the way
from there to there, you have 180 degrees. So this big angle right here
has got to be 60 degrees. This angle right here, it's
a complement of 30. So this is 60 degrees. And then this is 60 degrees. So this is an equilateral
triangle. All the sides are going
to be equal. So that side is equal to
that side, which is equal to this side. Now let me ask another
question. What is this side right here? Let me just draw the
shorter part. And once you memorize 30, 60,
90 triangles, you don't have to go through all this. But it's good to be able
to reprove it. So what's this length
right here? Well if we look right up
there, that's length a. This was just a mirror image,
so this is also length a. This whole base of the
equilateral triangle is 2a. Well it's an equilateral
triangle so all the sides are the same, so this is
2a and this is 2a. And just like that we were
able to figure out the hypotenuse. And they want to know
what the b is. Once we know two sides of a
right triangle, it's very easy to figure out the third side. So we know that a squared
plus b squared is equal to c squared. Let me write that down. Now what's a squared? a is 3 roots of 3. So a squared would be, let me
write it down, 3 roots of 3 squared plus b squared is equal
to, let me do it in another color. What's c? c, we just figured
out, is 2 times a. So it's 6 roots of 3 squared. That's what we did all this
stuff for, to figure out that this length is twice
that length. Fair enough. Now let's simplify. So if we take 3 roots of 3 to
the second power, that's the same thing as 3 squared times
square root of 3 squared. So that's 9 times 3
plus b squared is equal to 36 times 3. And so that's 27
plus b squared. 36 times 3 is 108. Subtract 27 from both sides. b squared is equal to 81. b is equal to 9. So that is choice a. Anyway, you should watch the
videos I've done in the trigonometry playlist on 30, 60,
90 triangles if you want to be able to do this faster. But I think that was useful. Because you've actually seem
how you can figure out the sides of a 30, 60, 90 triangle
without having memorized it ahead of time. Anyway, see you in
the next video.