Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. In a right triangle, if we know at least one angle and one side, we can find all the other given information using our concept of soka toa sine being opposite of our hypotenuse, cosine being adjacent over hypotenuse, tangent being opposite over adjacent. So if we're given A is 23.5 and we know that side B is 10, we can then find angle B because angle B is going to be complementary or 90° -, 23.5°. If we have 90° -, 23.5°, that's going to be 60 6.5° for angle B. Now angle C was already 90, so the three angles have to add up to 100 or 180, and they do. Now to find side A, we would know that tangent of 23.5 would equal the opposite over the adjacent. So A would just be 10 tangent 23.5° and we could grab our calculators and plug that in to get an approximation. If we wanted our sine C, we traditionally start with what's given. So we know that cosine of 23.5° would be 10 / C So if we wanted to solve for C, we'd get C equaling 10 divided by cosine 23.5°. And then we found our unknown angle and our two unknown sides. We would just plug those into our calculator and round to whatever the directions ask us. We also can solve all the information if we're given two sides of a triangle, two sides of a triangle. So on this two sides of a triangle we knew that side A was 14.5 and we knew that side B is 24.1. We can find side C by Pythagorean theorem, so 14.5 ^2 + 24.1 ^2, and then we would square root that to get the distance for C and grab our calculator at that point and plug it in. We could also find A by knowing that the tangent of angle A is the opposite over the adjacent. So side A would be tangent or angle A Sorry, angle A would be tangent inverse of 14.5 / 24.1. Plugging that into our calculator, once we find A, we can find B because B is going to be the complementary angle which is just 90 minus whatever the A was. We have application problems also. So if we have a hot air balloon rising vertically and the angle of elevation changes from 19.2° to 31.7°, the point on ground, on level ground is 125 feet from the point directly underneath the basket. So there's my hot air balloon. We want to figure out how how far to the nearest 10th of a foot does the balloon rise. So we're wanting this distance in here. So what we're going to do is we're going to set up two trig functions. Let's call this A and this B. So if we look at this base triangle, we'd have tangent of 19.2° equaling A / 125. If we look at the whole triangle, we'd have another equation that would say tangent of 19 point. No tangent of 31.7° = A + b that whole distance over 125. We want to solve for B. So if we're wanting to solve for B, we could take this top equation and know that A is going to equal 125 tangent 19.2°. So taking the second equation, if we cross multiplied, we get 125 tangent 31.7° equaling instead of A. We could put in 125 tangent 19.2° + B. To solve for B, we would just subtract, so we'd have 125 tangent 31.7° -, 125 tangent 19.2° equaling B. We could grab our calculators at this point to figure out how many feet the elevation has changed. We're going to have bearings. Bearings always measure. Bearings are always measured off the north-south line. Then we state the angle and then how Far East or West we went. So if we look at a couple examples, we always base it off the north-south line. So this one is going to be off the north line because it's above the east West line. We need to figure out the angle. So if we know that this is 30, we know that this is 60 because 30 + 60 gives us our right angle. So we're going to be N 60°. And in this case, we went to the east. So the bearing on this first example would be N 60 E. This next example, we're going to use the South line because we're below the east West line. 55° is here. So we know that 35° has to be here because the two angles have to add up to 90. So this is going to be South 35° W Now be careful. Sometimes they actually give you the angle you need. Say we have this, we'd be N 20° W in this example. We didn't have to find this being 70° because it already gave us the angle off the north-south line. Doing an example using application, a jet leaves a runway whose bearing is north 35 E so N 35 E after flying 5 miles. So they went 5 miles out here. The jet turns 90° on a bearing, so we're out here somewhere, and now the jet's turning 90° and flying S 55 E, so this is 55°. If it tells us we're flying, we've turned 90°. We know that this angle in here had to be 35 because 35 + 55 is going to give us a total of 90 and flies on a bearing of South for seven miles. So at that time, what is the bearing to the control tower? So what we're trying to figure out is this over here. So if we know that this original angle had to be a right angle, if that's 35, then this had to be 55 and we need to figure out how far we're going. So if we know that this is 90 and this portion's 55, we don't know this portion. Here a boat leaves a harbor on a bearing N 41 W. So there are north line and 41° W and travels 120 miles. How many miles north and how many miles West has the boat traveled? If we thought about drawing in a right triangle here, we'd have our north and our W coordinates. So we would know that the cosine of 41° adjacent over hypotenuse would give us 120 cosine 41° equaling how far north we've gone. We know that sine of 41° is opposite over hypotenuse, so our W would be 120 sine of 41°. That's our actual. If we wanted an approximate, we would grab our calculators, we would make sure we were in degree mode and we would type that in to get an approximate miles. Our next example talks about a jet leaves a runway whose bearing is north 35° E from the control tower after flying 5 miles. Then it the jet turns 90° and does a bearing of South 55° E and it travels 6 miles. And we want to know what's the bearing from the tower to where the jet now is. Well, if we know that this is a right triangle, because we turned 90°, we know that this is 35°. If we could figure out this Theta here, we'd be able to come up with our bearing because our bearing would be 35° plus whatever that Theta is. So looking at our right triangle, we know that we could use the tangent of our Theta to be 6 / 5 or Theta would be tangent inverse of 6 / 5. If we typed tangent inverse of 6 / 5 into our calculator, we'd get about 50°. So our bearing is going to be north 50 plus the 3585° E simple harmonic motion. The distance equal a cosine Omega T or distance equal a sine Omega T The D is the distance from the origin, so the maximum displacement is just our amplitude or the absolute value of A. The period of the motion is 2π over Omega where Omega is greater than 0. The period gives the time it takes for the motion to go through a cycle. The frequency is Omega over 2π where Omega is greater than 0. Or we could just think of it as 1 divided by the period. So if we have a distance or a simple harmonic formula of D equaling -5 cosine Pi 3T, our maximum displacement is going to be the absolute value of -5 or five. That negative actually tells me we're going down to start. The frequency is going to be R Omega in this case π thirds over R2 Pi. Well π thirds divided by 2π. That's the same thing as π thirds times 1 / 2π. So the pi's are going to cancel and we're going to get 1/6. The time required for one cycle is going to be 2π over Omega or 2π / π thirds. The pi's are going to cancel and 2 * 3 is 6. We would expect the time required for one cycle to be the reciprocal of the frequency, and six and one sixth are reciprocals of each other. A balloon, A ball on a string, A ball on a spring is pulled 7 inches below its natural position. The period of the motion is 4 seconds. Find the equation assuming simple harmonic motion. So we're going to start with D. It's going to equal -7 because it's going down. It's going to be a cosine graph because when we release it, we're releasing it from the down position. We're not releasing it from the the 00 position. So then we need to figure out if we know the period is 2π over Omega and that. Is going to be 4, so our Omega is 2π / 4 or Pi halves so it's going to be -7 cosine Pi halves T. Thank you and have a wonderful day.