50 foot tree casts a 40 foot shadow what is the angle at the top of the tree?

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- Nov 2nd 2009, 08:00 PMwenatcheemore triangles
50 foot tree casts a 40 foot shadow what is the angle at the top of the tree?

- Nov 2nd 2009, 08:10 PMjmedsy
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. - Nov 2nd 2009, 08:24 PMwenatchee
a little more detail i missed this whole section due to swine flu

- Nov 2nd 2009, 08:37 PMjmedsy
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. - Nov 2nd 2009, 08:54 PMwenatchee
that was a lot of help, i am going to try another to see if i can get it

- Nov 2nd 2009, 09:05 PMwenatchee
hight 9 length 12 r 15

sin= 12/15

cos= 9/15

tan= 12/9 - Nov 2nd 2009, 09:11 PMjmedsy
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 - Nov 2nd 2009, 09:20 PMwenatchee
I don't get why I would use the inverse

- Nov 2nd 2009, 09:26 PMwenatchee
so if the question is find sin A, and cos A would it be sin 15/12 and cos 5/9?

- Nov 2nd 2009, 09:29 PMjmedsy
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 - Nov 2nd 2009, 09:37 PMwenatchee
Okay, maybe if I sleep on it and start fresh in the morning when I am not so tired it will help. Thank you so much for your help. I think I am almost there.

- Nov 2nd 2009, 09:42 PMjmedsy
Consider this triangle:

http://www.andrewdothay.net/silverli...t-triangle.png

Remember "sohcahtoa",

Then: $\displaystyle tan(\theta)=a/b,sin(\theta)=a/c,cos(\theta)=b/c($

: $\displaystyle arctan(a/b)=\theta,arcsin(a/c)=\theta,arccos(b/c)=\theta$