Oblique traingles please help?

Question

when observed from a point P on the same horizontal level as the base, the angle of elevation of a spire is 39.6 degree and, when observed from a point 27.3 m above P, the angle of elevation is 30.9. find the height of the spire

Answer 1

The spire will be 27.3 m plus the height seen by the second observation.

The key I think is to use the Law of Sines

Draw out a picture of what's going on. What you can get is a oblique triangle. Using complimentary angles, you can find all 3 angles inside the triangles. You know the length of one side because its given.

The angle between the line made by Point P and the vertical is 90-39.6 = 50.4.
The angle between the line made above Point P and the vertical is 90+30.9 = 120.9.
The last angle between the lines made by point p and above point p is 180-120.9-50.4 = 8.7 deg

Then from law of sines

sin(8.7)/27.3 = sin(120.9)/L

solve for L = 154.8660 is the distance from point P to the top of the spire. Then the vertical height of the spire is

L*sin(39.6) = 98.7 m

Answer 2

x + 27.3 = height of spire
y= distance of point P from foot of spire
(x+27.3)/y = tan 39.6
x/y= tan30.9
(x+27.3)/tan39.6 = x/tan30.9
(x+27.3)/x= tan39.6/tan30.1
(x+27.3/ x=.76
x +27.3 = .76x
.24x = 27.3
x = 113.75
Spire height = 113.75+27.3 = 151.05m

Answer 3

i'll help u