The two methods of expressing bearing can be interpreted...

The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. $$(2,2)$$

Key Concepts

Trigonometric Angle Calculation

Trigonometry allows one to compute angles in a coordinate system by relating the sides of right-angled triangles. Techniques such as using the arctangent function to determine the angle of a point relative to an axis are crucial in converting between Cartesian coordinates and navigational bearings.

Rectangular Coordinate System

This fundamental system uses two perpendicular axes to define the location of points in a plane. It provides a framework where positions are given as ordered pairs, making it easier to apply algebraic and geometric methods to analyze spatial relationships.

Bearing Conventions

Bearings are used in navigation to indicate direction. They are typically expressed as angles measured clockwise from a reference direction, usually north. Understanding these conventions is essential to interpret and communicate the direction of objects relative to a fixed point.

Transcript

00:02 This question explores bering.

00:07 Bering is used in the navy in the air force, and it's used to express direction.

00:15 And it can actually be interpreted using normal x, y coordinates that most people are used to, in a map, let's say, for example.

00:25 And there's two ways of expressing bearing, and we're going to actually look at both of them here.

00:30 We're going to use an example.

00:33 So here on the left side, we have, a normal compass, right, the northeast, southwest.

00:40 And this is just the north section of our compass, right? it's just going to point upwards.

00:46 And let's say that we're going to have a point out a spot.

00:50 You're at the origin, right? and this is going to be zero, zero.

00:52 Origins always at zero.

00:55 And let's say you spot a boat out at three, four.

01:01 Right? so that's over three and up four.

01:03 So let's say that's somewhere over here.

01:07 This is going to be three four now that tells us right that this length is three and that tells us that this length here is four and the first way to find bearing is actually to find the angle from the north to the angle where you meet with the line or the point here right so the angle we're looking for specifically is going to be this one here now how do we find this angle? well, we have this, right? 3 -4 triangle here, so we can just find this angle here.

01:56 And since we know that the overall angle here is a right angle, right, between the north and x -axis, we can just subtract out here.

02:03 So we're going to use something that's called sokatoa.

02:12 It's extremely handy and i basically use it for all of my trigonometry questions.

02:19 This is going to be your sign, this is going to be your cosine, and this is going to be your going to be your tangent.

02:23 Here, o stands for opposite, h stands for hypotenuse, and a stands for adjacent.

02:30 And you basically choose an angle, and that will change your size, right? the only one that doesn't change is the hypothesis, the hypotenuse, right? and you want to take this with respect to the theta here.

02:45 So let's say that we have a 3 and a 4, right? so we have adjacent to this data, and we have an opposite to this data.

02:52 So that means that we're going to have to use tangent here.

02:55 Right.

02:56 So we're just going to do tangent of theta is equal to, again, opposite over adjacent.

03:20 Now plugging this into a calculator, right? so we have that this is going to be tan theta is equal to four thirds, right? opposite over adjacent.

03:29 So what we want to do is we actually want to take the tangent inverse of both sides.

03:34 And again, you will need a calculator for this.

03:37 So we take the tangent inverse of both sides.

03:40 This side will cancel out...

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