From a point P on level ground, the angle of elevation of he top of the tower is 26.83 degrees. From a point 25.0 meters closer to the tower and on the same line with P and the base of the tower, the angle of elevation of the top is 53.5 degrees. Approximate the height of the tower.
Think you can manage that? Try
secondly, how did you find out that P = 45.09 from the first triangle only?! you have one equation and two unknowns. There is no way you can solve for P like that! the answer has to have h in it.
make another attempt
i do not know what happened here. first, your -25 in the first line just disappeared into thin air. then, you assume that P = 25, which it doesn't.
So how do you find h?
There are 3 sides to a right-angled triangle and 3 angles.
One angle is 90 degrees and the other two are acute.
Sin, Cos and Tan give the ratio of 2 sides with respect to one of the acute angles.
You need to locate the sides "opposite to" and "adjacent to" an angle.
The longest side is the hypotenuse.
You can use SOHCAHTOA to help remember until you master it.
The small triangle is inside the big one.
The height "h" is the opposite to both angles and P is on the adjacent.
We use Tan if we are dealing with opposite and adjacent.
multiply both by P
divide by the Tan term for P
multiply both sides by (P-25)
Multiply out the left
Bring the P term and h together by subtracting h from both sides
add the 25Tan term to both sides
Use your result for h from the big triangle
multiply both sides by
Write this using h as a factor
Divide both sides by the factor of h
It's of course simplest to use your calculator to find the two values of Tan.
The big triangle gave us
This does not allow us to find h unless we know P.
Or... to find P, we need to know h.
Remember how simultaneous equations work?
If there are 2 things you do not know, then you need a 2nd clue.
For instance x+2=5, then you know x is 3 or you can work it out.
That's one clue needed to find one unknown.
But if we had x+y =5, we are stuck.
They could be 0 and 5, or 1 and 4, or 2 and 3, or -2 and 7 and so on.
When we have 2 unknowns, we need 2 clues.
In simultaneous equations, you may have just cancelled terms as follows.
add them... 2x+y-y=5+1=6 gets rid of y, so 2x=6, so x=3 and then y=2
Or.... we could write x=1+y from the 2nd equation,
then use this in the other
(1+y)+y=5, so 2y+1=5, so 2y=5-1=4, so y=2 and x=3
Do you see the idea ?
You may have covered these in class.
We use the same type of trick here.
H is the vertical height of both triangles.
The bases are different, one is P, the other is P-25
However we can use Tan(angle), using both angles given to write
out our two clues, since the sides we are dealing with involve "opposite"
We write P "in terms of h" using one triangle,
just as we did with the x and y values above.
Then we swop that into the 2nd equation, effectively getting rid of P
just as we got rid of x above to figure out what y is.
We got rid of P (wrote it in terms of h) because the question asked us to find h.
Study this idea, because it's an important maths technique.
You are only getting used to trigonometry,
so this problem contains quite a bit of calculations.
But it's important that you master the idea of cancelling one of 2 unknowns
if you only want one of them.