a lamp post casts a shadow of 8 feet long when the angle of elevation of the sun is 64 degrees. How tall is the lamp?

tan(angle) = opposite/adjacent

Here:  opposite = height; adjacent = 8ft;  angle = 64°,

So: tan(64°)=height/8  or height = 8*tan(64°) ft

$${\mathtt{height}} = {\mathtt{8}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{64}}^\circ\right)} \Rightarrow {\mathtt{height}} = {\mathtt{16.402\: \!430\: \!732\: \!632}}$$

height ≈ 16.4 ft

Well this one can be solved with the Side-Angle-Side theorem.

Assuming the lamp post is at a 90 degree angle from the ground we have this:

Where B is the height of the lamp post, A is the length of the shadow, and b is the elevation of the sun casting the shadow.

The sum of the angles in a triange always add up to 180 degrees, so since we know two of the angles, the third is easy. 180 - 64 - 90 = 26 (Assuming the lamp post is angled 90 degrees relative to the ground.)

So far we have a=26 b=64 c=90 and A=8 B=? C=?

So the law of sines states that A/sin(a) = B/sin(b) = C/sin(c), therefore:

8/sin(26) = B/sin(64) = C/sin(90) = 18.25. (Fun fact: sin(90) = 1)

So solving for C: C/sin(90) = C/1 = C = 18.25

So now we have a=26 b=64 c=90 and A=8 B=? C=18.25

now to solve for B:  B/sin(64) = 18.25    B=18.25*sin(64) = 16.4

Finally all the cards are on the table: a=26 b=64 c=90 and A=8 B=16.4 C=18.25

So to answer the question, the height of the lamp post is 16.4 feet.