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CA Geometry: Basic trigonometry

61-65, basic trigonometry. Created by Sal Khan.

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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