Some trigonometry problems- help needed (software assist allowed). Please show working out?

Question

1. From lookout #1 a fire is spotted on a bearing 050 deg. From lookout #2 the fire is seen on a bearing 020 deg. Lookout #2 is 10 km from lookout #1 on a bearing 120 deg. Assuming that the fire and the two lookouts are all on the same horizontal level find how far the fire is from each lookout?




2. A tower stands vertically at the base of a hill that inclines upwards at 30 deg to the horizontal. From a point 25 meters from the base of the tower and directly up the hill the tower subtends an angle of 52 deg. Find the height of the tower giving your answer correct to the nearest meter.




3. Points A,B and C lie on horizontal ground. From A the bearings of B and C are 330 deg and 018 deg respectively. A vertical tower of height 40 meters has its base at A. From B and C the angles of elevation of the top of the tower are 20 deg and 12 deg respectively. How far is B from C, to the nearest meter?

Answer 1

Preliminaries:
if you have planar triangle with vertices A, B, C and angles alpha, beta, gama (closed to these vertices respectively) and sides a, b, c (opposite of vertices respectively) then you have:

b sin(alpha) = a sin(beta)
b cos(alpha) + a cos(beta) = c.

It follows:

a = c sin(alpha)/sin(alpha+beta); b = c sin(beta)/sin(alpha+beta);
or: a/sin(alpha) = b/sin(beta) = c/sin(gama)that is known as sin theorem for planar triangle.

Using this we solve your problems.

1. If we denote lookout #1 as A and #2 as B we can get a triangle with alpha = 70 degrees, beta = 40 degrees, and c = 10 km. From above formulas we get: a = 9.994208 km (fire distance from lookout #1), and b = 6.8351 (fire distance from lookout #2).

2. Your triangle A (base of a tower), B (25 meters from tower up the hill), and C (tor of the tower) has angles alpha = 30 degrees, and beta = 52 degrees with side c = 25 meters. Therefore the height of the tower b = 25 sin(52)/sin(82) = 19.88875 = 20 meters.

3. You have alpha = 48 degrees. From tower height we can get b = 40 cotang(12) = 188.2516 and c = 40 cotang(20) = 109.9277 and then using cosine theorem for planar triangle a^2 = b^2 + c^2 - 2bc cos(alpha) = 140.768 m or 141 m.