A friend walks away from you a distance of 550 m, and then turns (as if on a dime) an unknown a

step 1 Once again, welcome to a new problem. Let's think about trigonometric ratios now. Let's think ab

A friend walks away from you a distance of 550 m, and then...

Key Concepts

Triangle Geometry

Triangle geometry involves the study of the properties and relationships between the angles and sides of triangles. In problems where positions, distances, and turns are involved, mapping the situation to a triangle framework allows one to use geometric relationships, such as those provided by the Law of Cosines, to determine unknown quantities.

Ambiguous Case of Trigonometric Functions

When using trigonometric functions like cosine, one must be aware of the ambiguous nature of the inverse cosine function. Since the cosine of an angle is the same as the cosine of its supplement, computations can sometimes yield two possible angles (an acute and an obtuse solution). This ambiguity is an important consideration when interpreting results from the Law of Cosines in real-world problems.

Law of Cosines

The Law of Cosines is a fundamental principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful when dealing with non-right triangles and allows one to calculate an unknown angle when all three side lengths are known, or an unknown side when two sides and the included angle are known.

Transcript

00:02 Once again, welcome to a new problem.

00:07 Let's think about trigonometric ratios now.

00:10 Let's think about triangle.

00:13 So let's say we have a triangle like this.

00:16 This is a, this is b, and this is c.

00:19 And then the opposite angle to side a, small a, is capital a.

00:26 Side b is capital b, and then side c is capital c.

00:31 So if you think about the cosine rule, the cosine rule says that if i wanted to get a squared, a squared is the same as the two opposite sides.

00:51 So b squared plus c squared minus 2 bc, cosine, cosine of a.

01:03 Of 8.

01:04 So that's the angle that's opposite the side of a.

01:08 If i wanted to get b squared, i would say b squared is the same as a squared plus c squared minus 2ac, cosine of capital b.

01:23 And then c squared is the same as a squared plus b squared minus 2ab, cosine of c.

01:32 So, cosine of c.

01:35 So these are the angles you're dealing with and we could always think about these sides as displacement.

01:46 So these are the sides that we're looking at as displacement.

01:51 So we have a problem here and in this particular problem, assume that you have a friend that's walking in a specific direction and we're assuming that they're going upwards a distance a so they're going upwards a distance a and then the distance a happens to be of 550 meters and then they make a turn and the angle that the turn is is a known so let's assume that this angle right here that they make a turn on, they travel distance b and assume b is 178 meters.

02:45 And this angle right here is theta, so that angle is theta.

02:53 And then finally what's going to happen is that there is a gap between their original position and their final position.

03:02 And that gap happens to be c and the distance c is the same as 432 meters that's the distance c that you're seeing right there this angle right here let's call it alpha and then this angle right here let's call it gamma so those are the angles this is your y direction and that's your x direction and and so the goal of this particular problem is to determine the angle between the initial direction and the direction of his final position.

04:14 Of the final position so that's what you're looking at or that's what you're looking at in this particular problem in terms of visuals that we're seeing the angle that we're looking at is so we're looking for for that angle so the time in the angle between his initial direction and the direction of the final position and that angle is going to be the alpha so that's alpha, that's this angle right here.

04:49 And then the second angle that you're looking for is by how much n degrees did his direction change? or what you can say is what's the total turn in angular direction? direction what's the total turning angular direction and by that we mean the gamma so what's what's the gamma we're asking for the gamma we'll jump in and go right away to the problem and what's going to happen is that we can we can come up with with the relationships so we know besides a b and c the first one is what looking for site b so we're going to say that b squared equals to b squared equals to a squared plus plus c squared minus 2 a c um cosine of alpha cosine of alpha so that's which you're seeing right there and then we're also saying also saying that we're looking at this other side, c squared, equals to a squared plus b squared, a squared plus b squared.

06:50 These are the two sides that you're looking at a and b.

06:54 We have theta minus 2ab cosine of data.

07:00 And then we are also saying that the two angles that you're seeing right here happen to be at a point.

07:07 So gamma plus theta, gamma plus theta equals to 180.

07:17 So those are the angles that you're doing.

07:21 And so we're going to say that we're going to say that we're looking for specific angles.

07:32 So this is what's going to happen.

07:37 We have three sides.

07:39 We have side a, side b, and side...