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effort
Senior Member

USA
38 Posts

Posted - 05/03/2013 :  10:49:44  Show Profile  Reply with Quote
I have to find the exact value of some inverse trigonometric funstions. I have the answers but don't know how to get the answers. If you will show me how to do one of the problems, I'm sure I can use that as an example to figure out the other 5 problems.


The exact value of the trig function tan((1/2)Arccos (5/13)) is (2/3) but I don't know how to get this answer. Will someone show me how to get this answer?
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someguy
Advanced Member

Canada
143 Posts

Posted - 05/03/2013 :  12:11:14  Show Profile  Reply with Quote
Hi Effort,

You are looking for tan(/2) where =arccos(5/13).

Using the half-angle identities for the tangent function is useful here.

tan(/2) = sin()/(1+cos()) = (1-cos())/sin()

Since arcos(5/13) is an angle between 0 and 90 degrees, we can just draw a right triangle, call one of the angles that is less than 90 degrees , and then label the lengths of the adjacent side and hypotenuse as 5 and 13 (so that cos=5/13). The Pythagorean theorem can be used to find the length of the third side.
Use this triangle with the identity tan(/2) = sin()/(1+cos()) to find the value of tan(/2).
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the_hill1962
Advanced Member

USA
1469 Posts

Posted - 05/03/2013 :  18:00:36  Show Profile  Reply with Quote
Quick response, someguy!
I was looking up the half-angle identity and started to work it out. Before I had finished, I logged back in to start a reply in and saw that you already have it all typed out!
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effort
Senior Member

USA
38 Posts

Posted - 05/06/2013 :  08:16:47  Show Profile  Reply with Quote
Here is my work: (1-(5/13)) / (12/13)

which using = (8/13)/(12/13) and using same change flip gives 2/3
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