A 40 ft pole casts a shadow

A).Express the angle of elevation θ of the sun as a function of the length s of the shadow
B).Find the angle θ of elevation of the sun when the shadow is 40 ft long. (Give your answer correct to 4 decimal places.)

A) arctan (40/s)

B) 45 degrees

Think of a right triangle with hypotenuse slanting to the right forming the angle θ with the base. This is the same as the angle of the sun in the sky.

The side opposite θ is the pole. The side adjacent to θ is the shadow.

A)

So if tan θ = opposite side/adjacent side, then

θ = arctan(opposite side/adjacent side) = arctan(40/shadow)

B)

If the shadow is 40 feet long the tan θ = 40/40 = 1
so

θ = arctan (1) = PI/4 = 45 degrees.

The pole is 40 ft. ; the shadow is s along the ground. The hypotenuse woudl be sqrt (1600 +s^2).
The angle would be arcsin[ 40/(sqrt (1600 +s^2))]
if the shadow is 40 feet long; this is arcsin 40/sqrt (1600 +1600)
arcsin 40/40sqrt 2
which is pi/4, I think. This seems right but I'm not sure..