CA Geometry: Basic Trigonometry 61-65, basic trigonometry
CA Geometry: Basic Trigonometry
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- 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
- 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.
- 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.
- 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.
- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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