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

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 $A$ 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 $\mathrm{C} ?$

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful in problems where two sides and the included angle are known, allowing the calculation of the third side, which is essential in solving many triangulation problems.

Triangle Geometry

Triangle geometry deals with the study of triangles, focusing on the relationships between sides and angles. Understanding these relationships, including properties like angle sums and congruence, is crucial for analyzing and solving problems that involve finding unknown distances or angles in planar figures.

Coordinate System Conversions

Converting between coordinate systems, such as from polar (using a distance and an angle) to Cartesian coordinates (using x and y components), is an essential technique. This conversion simplifies complex spatial problems by allowing the use of algebraic methods to compute distances and angles between points.

Bearings and Angle Conversions

Bearings provide a way to specify direction relative to a fixed cardinal point, typically north. Understanding how to interpret and convert bearings (like 'north of west' or 'north of east') into standard angles in a Cartesian coordinate system is vital for accurately mapping positions and solving navigation or displacement problems.

Transcript

00:01 Now here what we are given that there are three deers a, b and c that are grazing in the field.

00:06 Deer b is located 62 meters from deer a.

00:09 So i have taken deer a at the center, if you see here.

00:13 And b is at a distance of 62 meters from a at an angle of 51 degrees north of west, right? so this is the direction of north, here you have got west, so it's sort of like this, 51 degrees.

00:23 And deer c is located 77 degrees north of east relative to deer a.

00:27 So this is east and i've taken it like this.

00:30 And the distance between the deer b and c is 95 meters.

00:34 Now based on this you have to calculate the distance between a and c, right? so what i do is i take small a to be the distance between a and c, right? and based on this you can have, let's say you take a to be the origin and you have to find the coordinates of b and c, right? so c will be having coordinates, it's a cos of 77 degrees, a sin of 77 degrees, no? so this is the coordinate of c and for b the coordinate will be minus 62 cos of 51 degrees because it's along the negative x direction, right? or towards the west and north is positive.

01:18 So 62 sin of 51 degrees, right? or let's equate this a bit, simplify this rather.

01:31 62 cos 51 gives you a value of minus 39 and this is 62 sin 51, 48 .2, right? now what you can do is you can take the distance between d to be b and c or what i can do is i can write bc that will be equal to root over of a cos 77 minus of minus 39, so plus 39 squared plus a of a sin 77 minus 48 .2 squared...

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