Applied Trigonometry, Deduce Length of Segment in new triangle.

[max0005]

Homework Statement

When the sun is 74° above the horizontal a vertical flag pole casts an 8.5m shadow on the horizontal. Find the shadow cast when the sun lowers to 62° above the horizontal.

Homework Equations

The Attempt at a Solution

I drew the following diagram:

[PLAIN]http://img703.imageshack.us/img703/8046/sunq.jpg

(The point without label should be point E).

Hypothesizing AE to be a right angled degree I came up with the following:

AE:BD=AC:BC

BD and BE are the only two values which will remain constant in the 62° triangle.

However, I do not understand how to proceed. I am given only one side and one angle (excluding right angles) and the triangle on which I have to focus is a non-right triangle, hence sine, cosine and tangent cannot be used.

Sine and cosine rules require more data to be given, and it is impossible to figure out lengths of the sides now knowing at least the length of one side.

Suggestions?

 

[verty]

How large do you estimate angle BCD to be?

 

[max0005]

Less than 90 (DBC is right)... However, I need exact answers, not estimates. :(

 

[Mark44]

Your picture is well-done, but incorrect. When the sun's angle of elevation is 74 deg., the shadow cast by the flagpole is 8.5 m. From this information, angle BCD is 74 deg. The unknown here is the height of the flagpole, which we can call h.

Write an equation involving the angle (74 deg.), the length of the shadow, and h, and solve for h.

Draw a new triangle with the angle of elevation of the sun now at 62 deg. Write a new equation involving this angle, the length of the shadow and the height of the flagpole, and solve for the length of the shadow.

 

[max0005]

Hi Mark44, thanks for your answer!

I think I understood what you mean, but still have a question.

By putting angle BCD as 74° it means that the sun is (theoretically speaking) orbiting around that point. Being that point also the length of the shadow, it would that when the angle is 62° (Therefore the shadow longer.) point C would be further away from the flag's pole, therefore the center of the "orbiting cirlce" placed at a different point. Is my reasoning correct?

 

[Mark44]

Hi Mark44, thanks for your answer!

I think I understood what you mean, but still have a question.

By putting angle BCD as 74° it means that the sun is (theoretically speaking) orbiting around that point.

Around what point? I don't think it's useful to think in these terms. Instead think about a straight line that extends from the sun to the top of the flagpole, and continue it until it intersects the ground. The only points that are fixed here are the two endpoints of the flagpole, line segment BD. As the sun drops in the sky, point C moves farther away from the flagpole.

Being that point also the length of the shadow, it would that when the angle is 62° (Therefore the shadow longer.) point C would be further away from the flag's pole

There seem to be some words missing here, but yes, point C is farther from the base of the flagpole.

, therefore the center of the "orbiting cirlce" placed at a different point. Is my reasoning correct?

Again, I don't see any point in thinking in terms of an orbiting circle.