SOLUTION: A surveyor at point A, measures the horizontal angle between a marker at point B (Further along his/her side of the river) and a tower on the opposite side of the river as 62 degre

Algebra ->  Trigonometry-basics -> SOLUTION: A surveyor at point A, measures the horizontal angle between a marker at point B (Further along his/her side of the river) and a tower on the opposite side of the river as 62 degre      Log On


   

Question 1151057: A surveyor at point A, measures the horizontal angle between a marker at point B (Further along his/her side of the river) and a tower on the opposite side of the river as 62 degrees. He/she walks 50m directly to point B where he/she measures the angle between point A and the tower to be 45 degrees.
Draw a clearly labelled diagram showing this information and use it to calculate the distances between the tower and the two observation points.

Found 2 solutions by jim_thompson5910, MathLover1:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Diagram:

Points A and B are on the same side of the river. Point C is the tower's location on the opposite side of the river.

Side lengths:
a = unknown (this side is opposite angle A)
b = unknown (this side is opposite angle B)
c = 50 (this side is opposite angle C)

Angles:
A = 62
B = 45
C = 73
Angles A and B are given in the instructions.
Angle C can be found by solving A+B+C = 180.
Equivalently, C = 180-A-B.
Note how: A+B+C = 62+45+73 = 180.

The goal is to find the side lengths a and b.

Use the law of sines to solve for side a
a/sin(A) = c/sin(C)
a/sin(62) = 50/sin(73)
a = sin(62)*50/sin(73)
a = 46.1645509631375
a = 46.16455

Use the law of sines to solve for side b
b/sin(B) = c/sin(C)
b/sin(45) = 50/sin(73)
b = sin(45)*50/sin(73)
b = 36.9707866021467
b = 36.97079

Here is the updated picture of the solved triangle

The phrase "solve a triangle" basically means "find all three angle measures and all three side lengths".

All side lengths are in meters. The values for a and b are approximate.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!


first find the length of x and h:
tan%2845%29=x%2Fh
h=x%2Ftan%2845%29......since tan%2845%29=1
h=x

tan%2862%29=%2850%2Bx%29%2Fh......since h=x

tan%2862%29=%2850%2Bx%29%2Fx
tan%2862%29=50%2Fx%2B1.........tan%2862%29≈1.880726465346332
1.880726465346332-1=50%2Fx
0.880726465346332=50%2Fx
x=50%2F0.880726465346332
x=56.77m

then, 50%2Bx=50%2B56.77=106.77m

=>h=56.77m

the distances between the tower and the two observation points:

the distance AC:

AC=sqrt%28106.77%5E2%2B56.77%5E2%29
AC=120.9m->the distance between the tower and the observation point A


BC=sqrt%2856.77%5E2%2B56.77%5E2%29
BC=80.3m->the distance between the tower and the observation point B