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# CA Geometry: Basic trigonometry

## Video transcript

We're on problem 61. It says the point minus 3,
2 lies on a circle whose equation is x plus 3 squared
plus y plus 1 squared is equal to r squared. Which of the following must be
the radius of the circle? So the way to think about it
is, is that this point satisfies this equation. Any point on the equation will
satisfy both sides of this equality sign. So all we have to do is
substitute the x and the y here and see what r squared
has to be equal to. So let's do that. If we substitute the
minus 3 in for x. You get minus 3, I just
substituted for the x. Plus 3 squared plus,
now y, y is 2. 2 plus 1 squared is equal
to r squared. Minus 3 plus 3, that's just 0. 0 squared is 0. And then 2 plus 1 squared. So 3 squared is equal
to r squared. You could say r squared is equal
to 9 and then r is equal to 3 because you can't have
a negative radius. We see immediately that
r is equal to 3. So all you have to
do is substitute the x and the y values. Because any point that satisfies
this equality is on the circle, defined
by this equation. They say this point is on the
circle, so you just have to substitute them in and
just solve for r. Problem 62. Looks like we're going to
do some trigonometry. In the figure, below if sine
of x is equal to 5 over 13 what are the cosine of x
and the tangent of x? And I don't know if you've seen
the basic trigonometry videos, you might want to. But a good mnemonic for
memorizing sine, cosine and tangent is SOHCAHTOA. And that means SOH is sine is
equal to to opposite over hypotenuse. Cosine is equal to adjacent
over hypotenuse. And I'll tell you what these
mean in a second. And tangent, you might have
guessed, is equal to opposite over adjacent. So what does that mean? What is all of this mnemonic? So just you might want to
remember SOHCAHTOA. Then you could break
it down like that. So if I took the sine
of this angle. That means the opposite side
of this angle over the hypotenuse is equal to the
sine of this angle. Let's call this the opposite. This is the hypotenuse. This is the adjacent side. Because it's adjacent
to the angle. This one is opposite,
hypotenuse, adjacent. So the sine of x is equal to,
we know from our mnemonic SOHCAHTOA, opposite
over hypotenuse. And they tell us that that
is equal to 5/13. So opposite over hypotenuse
is equal to 5/13. Now we know that that's just
the ratio between the two. So we don't know. This could be 10, this
could be 26. This could be 1 and this
could be 13/5. Who knows. That actually doesn't matter. That's what's neat about
trigonometry. It's all about the ratio. So let's just assume
that this is 5. That the opposite
is equal to 5. And the hypotenuse
is equal to 13. Let me pick a different color. This is a little nauseating. All right. So if the opposite soon. is 5
and the hypotenuse is 13, what would the adjacent
be equal to? We could use the Pythagorean
theorem. So we could say the
adjacent squared. A squared plus the
opposite squared. So plus 5 squared, plus 25. Is equal to 13 squared. 13 squared is 169. If you subtract 25 from both
sides of this equation, you get a squared is equal to 144. A is equal to 12. We don't know that a is
definitely equal to 12. But we know that the ratio
of the opposite to adjacent is 5 to 12. Because we just assumed that
the opposite is 5. Anyway, so they want to know
what are cosine of x and tangent of x. So CAH. SOHCAHTOA. Cosine of x is equal to the
adjacent over the hypotenuse. The adjacent is 12. Hypotenuse is 13. So it's equal to 12/13. That's the cosine of x. And the tangent of x is equal
to opposite over adjacent. TOA, opposite over adjacent. So opposite is 4,
adjacent is 12. Equal to 5/12. And let's see what
choice that is. That's choice A, cosine
of x is 12/13. Tangent of x is 5/12. Next question. Looks like they want us to learn
a lot of trigonometry and geometry. Which is good. This is getting you warmed
up for the trig. In the figure below, sine
of A is equal to 0.7. So let's call this angle a. They say what is the
length of AC? So we want to know that. Let's call that x. So SOHCAHTOA. SOH tells us that sine of some
angle, let's call that theta, is equal to the opposite
over the hypotenuse. So sine of A, in this example,
is going to be equal to the opposite, 21, over the
hypotenuse, over x. And they tell us that the sine
of A is equal to 0.7. So now we can just solve this
equation for x and we're done. Let's see. So if you multiply x times
both sides, you get 21 is equal to 0.7x. And you divide both
sides by 0.7. You get 21/0.7 is equal to x. 21 divided by 7 is 3. So 21 divided by 0.7 is 30. So x is equal to 30. And that's length AC. That's choice C. Next question. 64. Approximately how many feet
tall is the street light? OK, so we can use some
trigonometry here. So if we know this angle, and
they give us the all of the trig ratios for that angle,
we're trying to figure out the height. So if I write SOHCAHTOA, what
are we trying to figure? So they gave us the adjacent. This is adjacent to the angle,
it's right beside it. The height that we're
trying to figure out, this is the opposite. So if we can have a trig you
function that deals with the opposite and the adjacent. Well that's tangent. TOA. Tangent of any angle
is equal to the opposite over the adjacnet. In this case, tangent of 40
degrees is going to be equal to the opposite, the opposite is
h, that's what we're trying to solve for, over
the adjacent. The adjacent is 20 feet. OK, tangent of 40 degrees isn't
something that most people have memorized, that's OK
because they gave it to us. Tangent of 40 degrees is 0.84. So we get 0.84 is
equal to h/20. So we can multiply both sides of
that by 20 and we get h is equal to 20 times 0.84. And that is equal to 16.8. And that is choice C. Problem 65. Right triangle ABC is
pictured below. Which equation gives the
correct value for BC? So this is what they want
us to figure out. This is BC right there. OK, let's read them. Let me write SOHCAHTOA, I
actually do this a lot. OK, so they're saying that the
sine of 32 degrees is equal to BC over 8.2. Is that right? Sine is opposite over
hypotenuse. BC is definitely the opposite. 8.2 is not the hypotenuse,
10.6 is the hypotenuse. So they're doing, this
is the adjacent. So this is wrong. So this should be a tangent. Tangent of 32 is equal
to BC over 8.2. This is the adjacent side,
adjacent to 32 degrees. That's the opposite, and
that's the hypotenuse. So that's not right. Choice B, cosine of 32. Cosine is adjacent
over hypotenuse. So cosine of 32 should
be 8.2/10.6. So this should be an 8.2 here. So this isn't right. OK, so the next one, tangent
of 58 degrees. Where are they getting
that 58? Well, they know that this is
32, this is 90, so this is going to be 180 minus
32 minus 90. Because the angles in a triangle
add up to 180. So this angle right
here is 58. And now if we use that angle,
we have to relabel opposite and adjacent and all that. So from this angle's point
of view, tangent is opposite over adjacent. So if we write the tangent of
58 is equal to the opposite side, should be equal
to 8.2 over its adjacent side, over BC. This is adjacent
to this angle. It was opposite this
angle, but BC is adjacent to this angle. So that's what they wrote. So choice C is correct. And we're done. I'll see you in the
next video. Well we're not done with
the whole thing. I'm done with this video. See