# trig

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as the sun rises it casts a long shadow over a boy who is leaning next to a tree. the tree is 35 feet tall and the sun's rays create an angle of depression of 70 degrees with the top of the tree. how long is the shadow

• trig -

Forget about the boy. Not relevant to the solution.

So, now we have a triangle with one leg 35, and the angle at the top of the tree is (90-70)° = 20°.

So, if the shadow is s feet long, then

s/35 = tan 20°
s = 12.7 ft

Looks like a short shadow to me. A smaller angle of depression would make a longer shadow.

• trig -

I believe that the final step would be 35 divided by tan20. As Steve explained, the angle between the ground and the shadow is 20 (complementary to the 70) and TAN is opposite/adjacent or the

35m height of tree divided by the unknown length of shadow. Algebraically, you multiply both sides of the equation by 's' or whatever variable letter you want to use -- thus getting it out of the denominator.

It looks like 35m = (tan20) multiply 's'. In order to get the 's' on its own, you then need to divide both sides by TAN20.... 35m/(tan20) = s thus giving the previously unknown length of the shadow and retaining the unit of measurement, metres. about 96.16m. The smaller the angle from the ground, the greater the shadow.... when angle is 45, the ratio is 1:1. You'll note that directly overhead, 90 degrees, there is no shadow at all.

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