i need help witha math problem

Question

A surveyor is 40m from the edge of a building. The angle of elevation from the surveyor to the top of the building is . What is the height of the building?

Answer 1

Hello Had!

I hope I can help you today!

For word problems like this one I find it always helps to draw a picture.

But first, what do we know?

The surveyor is 40 m from a building, and there is an angle of 55 degrees from the surveyor to the top of the building.

What does this mean? Well, the angle refers to the angle between the ground and the line going from where the surveyor is standing to the top of the building.

Let's draw a picture:

/ |

/ |

/ | height = ?

/ |

o / |

T/_55°__| building is over here

40m

The T with the O for a head is the surveyor.

So we can construct a right-triangle using the information given to us in the problem.

We know the bottom length of the triangle is 40 meters.

And we have the angle.

But we want to know the height of the building, which corresponds to the length of the opposite side of the right triangle. (the side opposite the 55 degree angle).

Do you know any trigonometric functions which incorporate these 3 values?

The angle, the adjacent side, and the opposite side?

If you recall the definitions of cos, and sin, and tangent, we have:

cos(theta) = adj / hypotenuse

But we don't know the hypotenuse, so this isn't the best option.

sin(theta) = opp / hypotenuse

Once again, not the best.

tan(theta) = opp / adj

Yes!

We know adj, and theta, so the only thing in this equation we do not know is the length of the opposite side!

This means we can solve for it!

tan(theta) = opp / adj

adj * tan(theta) = opp

Thus the height of the building is 40 * tan(55).

(make sure your calculator is set to use degrees, not radians btw)

In general, problems that involve right triangles often can be solved by manipulating these definitions of cos, sin, and tangent.

For example:

hypotenuse * cos(theta) = adj

hypotenuse * sin(theta) = opp

adj * tan(theta) = opp

If we know 2 out of 3 of the values in any of these equations we can always solve for the others.

We could even solve for the angle using the inverse trigonometric functions.

For example, if we knew the height of the building and how far away the surveyor was, and were asked what the angle between the surveyor and the top of the building was, we could solve:

tan(theta) = opp / adj

then taking arctan of both sides (arctan = tan inverse)

arctan(tan(theta)) = arctan (opp / adj)

inverses cancel the original function so:

theta = arctan (opp/adj)

And then we'd have the angle!

(You do need to be careful about what quadrant the angle is in though. as we could have opp and adj both being positive, or both being negative, or one being negative, and that will tell you what quadrant the angle actually should be in. You can add pi I believe to get the correct angle however, as tangent has a period of pi)

Anyway, I hope this helps! Let me know if anything here is confusing or if you have any questions!