Three towns A,B&C are located such that the distance between A&B is 15km and the distance between A&C is 90km, if the bearing of B from A is 075 and the bearing of C from A is 310.
(a) draw the diagram
(b) find the distance between B and C
(c) find the bearing of C from B
If A is at origin
Then C is 360-310 = 50 degrees west of north
and B is 75 degrees east of north
total angle CAB = 50 + 75 = 125 degrees
law of cosines
a^2 = b^2 + c^2- 2 b c cos 125
a^2 = 90^2 + 15^2 - 2 (90)(15)cos 125
solve for a
then use law of sines for other angles
if you find angle B for example you know B +75 + angle of BC to north south line is 180
I will assume you made the sketch
then you can see that the angle CAB = 125°
and we have a clear case of the cosine law.
BC^2 = 90^2 + 15^2 - 2(90)(15)cos 125°
= ....
Find BC
Once you have BC, use the sine law to find one of
the other angles. I suggest finding angle C to avoid any
conflicts with the ambiguous case.
Of course once you have 2 angles of the triangle, the third is routine.
Put all your angles in your sketch and you should be able to find the
bearing of C from B.
All angles are measured CW from +y-axis
b. BC = AC - AB = 90[310o] - 15[75o],
X = 90*sin310 - 15*sin75 = -83.4 km..
Y = 90*Cos310 - 15*Cos75 = 50.1 km.
BC = -83.4 + 50.1i = 97.3 km[-59o] = 97.3km[59o] W. of N. = 97.3km[301o].
c. Bearing from B to C = 301o(see part b).
To answer the question, let's start by drawing a diagram to better understand the given information.
(a) Drawing the diagram:
To create the diagram, follow these steps:
1. Draw three points representing towns A, B, and C.
2. Label the distance between A and B as 15km.
3. Label the distance between A and C as 90km.
4. Use bearings to add the angles to the diagram. The bearing of B from A is 075, which means we need to draw a line from A to B with an angle of 75 degrees clockwise from the north. Similarly, the bearing of C from A is 310, which means we need to draw a line from A to C with an angle of 310 degrees clockwise from the north.
The diagram should look like this:
```
A ----------------------- B
\
\
\
\
\
\
\
\
\
C
```
(b) Finding the distance between B and C:
To find the distance between B and C, we can use the cosine rule. The cosine rule states that in a triangle ABC, where sides a, b, and c are opposite angles A, B, and C, respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we need to find the distance between B and C, which is the side opposite angle A in triangle ABC. Let's label this distance as BC.
Using the cosine rule, we have:
BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(A)
Substituting the known values:
BC^2 = 15^2 + 90^2 - 2 * 15 * 90 * cos(75)
Calculating this equation will give you the square of the distance between B and C. To find the actual distance, take the square root of the result.
(c) Finding the bearing of C from B:
To find the bearing of C from B, we need to determine the angle that the line from B to C makes with the north direction.
Using basic trigonometry, we can find this angle by calculating the inverse tangent of the ratio of the horizontal and vertical distances between B and C. Let's label this angle as angle CBD.
tan(BDC) = (BC / AC)
Finally, to find the bearing of C from B, subtract angle CBD from the bearing of B from A (075 in this case) and adjust for any negative angles by adding 360 if needed.