Three deer, A, B, and C, are grazing in a field. Deer B is located 62 m from deer A at an angle

step 1 All right, welcome back. So today for our problem, we have three deer. We have deer A right here

Three deer, A, B, and C, are grazing in a field. Deer B is...

Three deer, $A, B,$ and $C,$ are grazing in a field. Deer $B$ is located 62 $\mathrm{m}$ from deer $\mathrm{A}$ at an angle of $51^{\circ}$ north of west. Deer $\mathrm{C}$ is located $77^{\circ}$ north of east relative to deer $\mathrm{A}$ . The distance between deer $\mathrm{B}$ and $\mathrm{C}$ is 95 $\mathrm{m}$ . What is the distance between deer $\mathrm{A}$ and $\mathrm{C} ?$ (Hint: Consider the law of cosines given in Appendix E.)

Key Concepts

Law of Cosines

This theorem relates the lengths of the sides of any triangle to the cosine of one of its angles. It is extremely useful in problems where two sides and the included angle, or all three sides, are known. The formula, c² = a² + b² - 2ab*cos(C), allows one to calculate an unknown side or angle in a triangle, making it invaluable for triangulation in two-dimensional geometry.

Angle Conversion

When directions are given using terms like 'north of east' or 'north of west,' it is necessary to convert these into standard geometric angles measured from a fixed reference, such as the positive x-axis. This ensures that trigonometric calculations are performed correctly and consistently, and is a key step in solving problems that involve directional bearings.

Triangle Geometry

Problems involving distances and angles in a field often form the vertices of a triangle. Understanding the properties of triangles, such as internal angle sums and side relationships, allows one to apply geometric theorems like the law of cosines effectively. This concept underpins the process of triangulation, where unknown distances or angles are inferred from known values.

Transcript

00:01 All right, welcome back.

00:02 So today for our problem, we have three deer.

00:05 We have deer a right here.

00:13 We've got deer b, an angle, that is due west right there, an angle of 51 degrees.

00:24 Deer c, an angle of 77 degrees above east of deer a.

00:41 All right, and we know this distance right here between deer a and deer b is 62 meters.

00:48 And between deer b and deer c is going to be 95 meters.

00:56 All right, and in this problem, we're trying to find this distance right here, the distance between deer a and deer c.

01:06 So we can set up an equation right here.

01:09 We're gonna use the law of cosines.

01:12 So if we, if this is angle b right here, we're gonna call this side angle, or side, this is angle c we're gonna call this side c and the side side a all right so just write that in just to be super clear it's not a whole lot of room on the side and a little c right there all right to love cosines one of the forms we can find it in is that b squared is equal to a squared plus c squared minus 2 a c cosine of angle b all right so to find um side b right here we have uh we're given sides a and c we need to find angle b so the first thing i would do looking at this problem is you can see this angle right here and we know that this right this right here is a flat line which means that the angles along it are going to add up to 180 degrees so we can say that 180 is equal to 51 plus 77 plus angle a and if you subtract the 51 and the 77 from both sides you'll find that a is equal to 52 degrees.

02:54 So i'll just write that in here, 52 degrees.

02:58 All right.

03:01 So that's still not the angle we're looking for quite yet.

03:05 We can use the law of signs to find this angle right here, angle b.

03:12 That's the angle we're really looking for.

03:17 And another way to find it would be to find this angle c.

03:20 And then subtract angle a and angle c from 180 degrees and that will give us angle b because there are 180 degrees in the triangle.

03:30 So we're going to do it that way because to use the lab signs, the formula for that is sign of an angle or the length of that angle is equal to sign of another angle or the length of that side.

03:48 All right.

03:51 And using this equation right here, we have all of these measurements except for sign of angle c...

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