Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Right triangle trigonometry. Sine as opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. Sometimes we remember this by Soka Toa. Soka Toa giving us the letter of each pairing. In a right triangle, we have what's called cofunction identities where we relate Theta to the other angle. That's not the right angle. So if one angle is Theta, the other angle has to be 90° minus Theta. Because the two angles are what are known as complementary angles, they add up to 90°. So drawing a right triangle and looking at the sides, sine Theta is b / C or opposite over hypotenuse. If we looked at 90° minus Theta, that would be really cosine for b / C So if we look here we have AB over C for our cosine of 90° minus Theta, but we also have AB over C for our sine Theta. So those two must be equal by the transitive property. Transitive property says if A equal B and b = C, then a = C Since these two equations each have AB over C, they must equal each other. So the cofunction identity is a sine Theta equal cosine of 90° minus Theta. Doing the same pairing for the other five trig functions, we get cosine Theta equaling sine of 90° minus Theta, tangent Theta equal cotangent of 90° minus Theta, cosecant Theta equaling secant of 90° minus Theta, secant Theta equaling cosecant of 90° minus Theta, cotangent Theta equaling tangent of 90° minus Theta. We also, in a right triangle by the Pythagorean theorem, know that a ^2 + b ^2 = C ^2, where C is the hypotenuse and A&B are the legs of the right triangle. We also know that the three angles have to add up to 180, but C is always 90°, so that leaves the A&B have to add up to 90°. Once again, those are called complementary. We're also going to talk about bearings. Bearings always start from the north-south line. So if we said we want north 12° E, what it says is starting with our compass concept, we start from the north line and we're going to go 12° to the east. Now, if we know that that's 12°, we can actually find this angle over here because the two angles have to add up to be 90. So if we needed this other angle, we could realize it's 78°. If we look at another one, say S, 54° West, this time we'd start on our S line. We'd go the West direction, 54°. If I needed this other angle, I know that these two angles here have to add up to give me a 90° angle, so that would be 36°. So now if we know anything in a right triangle, we ought to be able to find the other sides. So given A equal to and C = 7, if we draw a right triangle, if A is 2 and C is 7, we ought to be able to find side B, angle A, and angle B. So by Pythagorean theorem we know that 2 ^2 + b ^2 = 7 ^2, So 4 + b ^2 equal 49, B squared equal 45, B equal sqrt 45. We don't care about the negative because we're talking about a distance here. We could throw that into our calculators and get an approximate, but sqrt 45 is the exact answer. So we'd have tangent of a equaling b / 2. Now if we're going to use that B, we want to use the exact value of sqrt 45 SA equal tangent inverse of sqrt 45 over the given 2. Sometimes we actually go by only what's the given values. If we look at the given values, we could realize that cosine of A was 2 / 7. So we can find A easily by taking the cosine inverse of 2 / 7. Once we have A, we use the fact that we know that A+B equal 90, so B is just going to be 90° minus whatever we found for our A a moment ago. We'd use our calculator to get an approximate if we knew an angle and aside, we'd go through very similar procedure. Say we knew that A was 28° and we knew that C was seven, and we want to find little A, angle B and side B. Here once again, A+B had to equal 90°. So if we know angle A, we can figure out B by just 90 -, 28. 90 -, 28 is 62. And we would find little A and little B by using our trig. So we know that sine of 28 degrees would equal a / 7, so a is 7 sine 28°. We know that cosine cosine of 28° would equal b / 7, so we would put B is 7, cosine 28. It's always best, if possible to not use approximates, because approximates are going to compound errors. If we can use the original given information, that's always better. Once we had found our A, technically we could have done Pythagorean theorem, but we had already rounded A probably in our calculator and that would have given us B being more off from the actual answer than using that cosine or that trig function. Thank you and have a wonderful day. This is Anna Cox.