Answers:
- missing angle = 46.411 degrees
- total horizontal distance = 274.813 meters
Both values are approximate.
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Explanation:
h = height of the tower
We have two right triangles, one of which has the acute angle 38 degrees which is on the left side. Focus on this triangle to find h
sin(angle) = opposite/hypotenuse
sin(38) = h/200
200*sin(38) = h
h = 200*sin(38)
h = 123.132295 is the approximate height of the tower in meters
We'll use that tower height to find the other angle of elevation, which I'll call x for now. Focus on the triangle on the right.
sin(angle) = opposite/hypotenuse
sin(x) = h/170
sin(x) = 123.132295/170
sin(x) = 0.724308
x = arcsin(0.724308)
x = 46.411312
x = 46.411
The wire on the right side forms an angle of roughly 46.411 degrees. This is the angle made between the wire and the ground.
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So far, we've only used sine to figure out the tower height h and the missing acute angle along the ground. Turn to cosine to find the distances along the ground (aka the horizontal pieces of each triangle). I'll call those pieces y and z.
Start with the triangle on the left
cos(angle) = adjacent/hypotenuse
cos(38) = y/200
y = 200*cos(38)
y = 157.602151
Then move to the triangle on the right side
cos(angle) = adjacent/hypotenuse
cos(x) = z/170
cos(46.411312) = z/170
z = 170*cos(46.411312)
z = 117.211014
Lastly, add up the values of y and z to get the distance between the cables along the ground: y+z = 157.602151+117.211014 = 274.813165
When rounding to three decimal places, we get 274.813 meters
