50 foot tree casts a 40 foot shadow what is the angle at the top of the tree?
to find the angle formed by the ground and the hypotenuse, you calculate the inverse tangent of (50 / 40), 51.53 degrees. (remember, sohcahtoa)
Since a triangle has 180 degrees and the angle made by the ground and the tree is 90 degrees, the angle made by the tree and the hypotenuse is 180 - (90 + 51.54), which is 38.66 degrees. I'd be happy to go into more detail if you'd like.
You could also calculate the inverse tangent of (40/50). In this case, you would be dealing directly with the angle formed by the tree and the hypotenuse.
sure,
first ill cover "sohcahtoa."
this stands for:
"Sin:Opposite/Hypotenuse, Cosine: Adjacent/Hypotenuse, Tangent: Opposite/Adjacent"
for right triangles, you can do something like this:
Consider the right triangle made by the ground and the tree. To find the angle made by the tree and the hypotenuse of this triangle, first identify how the sides of the triangle relate to this angle. The adjacent side of this triangle is the tree, which is 50 feet, while the opposite side of this triangle is the length of the shadow, which is 40 feet. Look above at sohcahtoa. We can see that tangent of this triangle is opposite / adjacent. To find the desired angle here, we take inverse tan (40/50), which equals 38.66 degrees.
these are true for the angle which is formed by the height and the hypotenuse. Therefore, the inverse operations of those values (e.g. inverse tan (12/9), inverse cos (9/15), inverse sin (12/15)) yield the value of the angle itself
Use this: Right Triangle Angle And Side Calculator
consider the angle a which equals 38.66 degrees
also consider, its opposite side = 40 and its adjacent side = 50
this means that
$\displaystyle tan(a)=tan(38.66)=4/5=40/50
$
by definition of inverse, if $\displaystyle tan(38.66)=40/50$ then $\displaystyle arctan(40/50)=38.66$ degrees
note that arctan is the same as inverse tan