The bearing of B from A 12km is 038°.the distance between A and C is 17km.if C is on a bearing 152° from A find:(a) the bearing of A from C (b) the distance between point B and C (C) the bearing of C from B
(a) add 180°
(b) in triangle ABC, with the usual labeling of sides and angles, angle A = 114°, so using the law of cosines, that distance is
a^2 = 12^2 + 17^2 - 2*12*17 cos114°
(c) use the law of sines to find angle C:
sinC/c = ainA/a
Now you can use C to find the bearing.
All angles are measured CW from +y-axis.
Given: AB = 12km[38o], AC = 17km[152o]. BA = 12km[38+180].
a. CA: bearing(direction) = 152+180 = 332 deg.
b. BC = BA [218]+AC[152] = 12[218']+17[152]
BC = (12*sin218+17*sin152)+(12*cos218+17*cos152)i
BC = 0.593-24.5i.
BC = sqrt(0.593^2+24.5^2) = 24.51km[-1.4o] = 24.51km[1.4o] W. of N.
c. Bearing(direction) = 1.4 deg W. of N. = 358.6 deg CW.
To find the answers to these questions, we can use basic trigonometry and geometry concepts. Let's break it down step by step:
(a) To find the bearing of A from C, we need to subtract the given bearing of B from A (038°) from 180° because a straight line is a total of 180°. This is because the bearing of C from A will be the opposite direction of B from A.
Therefore, the bearing of A from C is 180° - 038° = 142°.
(b) To find the distance between points B and C, we can use the concept of applying the Law of Cosines to a triangle formed by points A, B, and C. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a is the distance between A and C (17 km), b is the distance between A and B (12 km), c is the distance between B and C (which we want to find), and C is the angle between sides a and b (180° - 152° = 28°).
Plugging in the known values:
c^2 = 12^2 + 17^2 - 2 * 12 * 17 * cos(28°)
c^2 = 144 + 289 - 408 * cos(28°)
c^2 ≈ 433.79
To find c, take the square root of both sides:
c ≈ √433.79
c ≈ 20.83 km
Therefore, the distance between points B and C is approximately 20.83 km.
(c) To find the bearing of C from B, we subtract the given bearing of B from A (038°) from 180°. This is because the bearing of C from B will be in the opposite direction of B from A.
Therefore, the bearing of C from B is 180° - 038° = 142°.
In summary:
(a) The bearing of A from C is 142°.
(b) The distance between points B and C is approximately 20.83 km.
(c) The bearing of C from B is 142°.