Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Right triangle trigonometry. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent or sine over cosine. Because the hypotenuses would cancel, sometimes we think of ways to help us memorize this, and so CAH TOA is sometimes the way students used to remember. Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. We also have reciprocal functions. Cosecant Theta is just one divided by sine. Secant Theta is 1 divided by cosine and cotangent Theta is 1 divided by tangent or cosine divided by sine. When we look at a right triangle, we have a right angle and we have some given angle Theta. If we look at this triangle labeled AB and C for our sides, our cosine is going to be the adjacent side over the hypotenuse, or B / C The adjacent side is the side that makes up the angle, so it's attached to our given angle here, sine Theta. The opposite side is the side that doesn't attach to the angle or doesn't make up one of the two sides of the angle. So A / C the hypotenuse is always the side opposite the right angle, and it's also always the bigger side. Tangent a / b secant is just the inverse of cosine. So C / b cosecant just the inverse of sine C / a cotangent the inverse of tangent b / a. Now, if we know that this is Theta, we know that the angle up here has got to be 90° minus Theta or Pi halves minus Theta because we know that the three angles of a triangle, Theta plus 90 plus the unknown has to equal 180. So that unknown, if we thought about subtracting everything to the other side, we'd subtract Theta, we'd subtract 9180 -, 90 is going to be 90. And then the Theta we can't combine because they're not like terms. So 90 minus Theta is that third angle. So we can now do all of our trig functions on that third angle. What's the cosine of 90° minus Theta? Well, the cosine was the adjacent sine over the hypotenuse, or a / C. What's the sine of 90° minus Theta? Well, sine is opposite over hypotenuse, so b / C What's the tangent of 90° minus Theta? The tangent is the opposite over adjacent b / A. Now we can find the other three trig functions just because they're inverses. Secant is the inverse, the reciprocal of cosine. So the reciprocal of cosine is C / A, cosecant is the reciprocal of sine C / b and cotangent is the reciprocal of tangent or a / b. Now, if we have b / C here and b / C here by the transitive property of mathematics, which says if A equal B and b = C, then a = C, that's called the transitive property. By the transitive property, we're going to come up with things that are called cofunction identities, and the cofunction identities are going to say this b / C and that b / C equal. So our cosine Theta has to equal our sine of 90° minus Theta. If we look here we have a / C and A / C so those two have to be equal. So sine of Theta is going to equal cosine of 90° minus Theta. Continuing in this pattern, A / b here equals A / b here. Hence tangent Theta has got to equal the cotangent of 90 minus Theta C / b here, C / b here. So secant Theta is going to equal cosecant of 90° minus Theta. Cosecant Theta has got to equal secant of 90° minus Theta because those were both C / A. And finally cotangent Theta is going to equal tangent of 90° minus Theta because those both equaled B / A. Given a right triangle with an unknown angle Theta and two sides 12 and 20, we want to find our third side by using Pythagorean theorem that says a ^2 + b ^2 = C ^2. We know that if A was 12 and we don't know B&C is 20, we can find B. So b ^2 is 20 ^2 which is 400 -, 12 ^2 which is 144. So b ^2 is going to be 256 and 256 square root is going to be 16. So this unside would unknown side would be 16. Now if we wanted to come up with all six of our trig functions, cosine Theta would be adjacent over hypotenuse 12 twentieths which would reduce to 3/5. Sine Theta would be 16, the opposite side over the hypotenuse which would reduce to 4/5. Tangent we can think of as sine over cosine or the opposite over adjacent, so 16 twelfths which reduces to 4 thirds. Secant Theta is just the reciprocal of cosine, cosecant Theta is just the reciprocal of sine, and cotangent Theta is the reciprocal of tangent. Now if we actually want to define the angle, we could use any of these six trig functions. Let's say cosine Theta equal 3/5. So Theta would equal cosine inverse of 3/5. So if we looked at our calculator and we said cosine inverse of 3 / 5, second entry, oh, cosine inverse, sorry, 3 / 5, let's not do second, let's just hit enter. So about 53.13°. Make sure you're in degree mode. So if you click on your mode button, you'd make sure you're in degrees. Quit out of that. So 53° approximately. So theta's approximately 53°. So what would the other angle up here be? That angle had to be 90 minus Theta, 90° minus Theta. So the other angle would be approximately 4037 if we have cosine PI6, tangent Pi, force minus sine π thirds. We can get all of this by knowing our unit circle, but we can also get it off of some right triangles in case we don't know our unit circle. If we thought about a 4545 right triangle, we would know that these sides have to be the same. And if we were referring back to our unit circle concept, we know the hypotenuse is always one, so we'd get a ^2 + a ^2 equal 1 ^2 2, A squared equal 1A squared equal 1/2, so a = sqrt 1/2 positive and negative. When we rationalize that we'd get root 2 / 2. Now, because we're referring to a distance, we're actually going to just use the positive root 2 / 2. So when we look at this triangle, tangent of π force or π force is the same thing as 45°. So the tangent is just opposite over adjacent. And this case it would be root 2 / 2 over root 2 / 2 which is really just one if we looked at a 3060 right triangle. Think about and a unit circle. The hypotenuse is always one. If I dropped this other half to make it into an equilateral triangle, we would end up having this being a 30° also and an equilateral triangle. The sides have to be the same. So if this whole base is 1, then half the base is 1/2 and now we could find this side, let's call it B right triangle. So we know that 1/2 ^2 plus b ^2 would equal 1 ^2. So b ^2 is going to equal 1 -, 1/4 or B is going to be b ^2 is going to be 3/4. If we square root it, we would get B equaling sqrt 3 / 2. Technically positive and negative, but because we're going to do a distance, we're going to use the positive. So remembering that Pi 6 is really 30° and π thirds is really 60° off of this triangle, we can find our cosine of Pi 6. So our cosine of 30° right here is going to be the adjacent side, root 3 / 2 over the hypotenuse. So root 3 / 2 / 1 or just root 3 / 2. These were getting multiplied together by the tangent Pi force. And then we're going to subtract sine of π thirds. Well, here is my π thirds. My 60° sine is the opposite side over the hypotenuse, so root 3 / 2 again over 1. And in this case root 3 / 2 * 1 minus root 3 / 2 is going to give us a final answer of 0. So this whole problem equals 0. If we wanted to write cofunction identities, cosine is the cofunction identity of sine of 90° -, 37°, so this one's going to be sine of 53°. The cosine of 37 is the same thing as the sine of 53 secant of five Pi twelves. That's going to be cosecant of π halves -5 Pi twelves. Remember, we need to use the same unit, so for in radians we use radians, for in degrees we use degrees. Pi halves is the same thing as six Pi twelfths. If we have 6 Pi twelfths and we subtract 5 Pi twelfths, we're going to get a single Pi 12th. Here tangent is the same thing as cotangent of π halves -2π ninths. So now we need to get a common denominator and that's going to be eighteenths. So 9 Pi eighteenths -4 Pi eighteenths. So tangent of 2π ninths is the same thing as cotangent of five Pi eighteenths. We're given a triangle and we know 1 angle and one side. First thing we can do is we can always find this other angle 90° -, 42°. So this unknown angle has got to be 48° because they all have to add up to be 180, all three angles. So 48 and 42 is 90 + 90. Now if we wanted to find A and C, we know that we want to find the. If we started with our 42° angle, we want to find the opposite side and we know the adjacent side. So if we want to find the opposite and we know the adjacent, we're going to use tangent. So the tangent of 42 is opposite over adjacent. So we'd get 16 tangent 42° equaling our A. We can grab our calculators and plug that in. If we wanted to know our C We want the hypotenuse over the adjacent. The hypotenuse over the adjacent is really just our secant of 42. If we think about secant as being one over cosine, when we put this into our calculator, we'd get 16 equaling cosine of sorry C equaling 16 divided by cosine of 42. If we know cotangent Theta is 5 sevenths, then tangent of π halves minus Theta, that's a cofunction identity. So if we have Theta here and we have π halves minus Theta here, and cotangent and tangent are cofunctions, it's just going to equal 5 sevenths by the transitive identity. The last one, if we have some application problems, let's say that we have a tree and we want to know how high the tree is based off of the sun being up here somewhere. If the sun's up here and we look from this location and we see a 40° angle and the tree base we can measure is 26 feet away, we can actually find the height of the tree. This would be tangent 40 would equal H / 26 or the height of the tree would be 26 tangent 40. If we had a a truck say driving along a very steep Rd. A road with a 12% incline. The truck went 1000 feet and we want to know how high the truck is from when it started, how much elevation had changed, what was the increase in the altitude? In this case we would use sine of 12° because we're trying to find the opposite side and we know the hypotenuse. So a would equal 1000 times sine of 12°. Thank you and have a wonderful day.