Author 
Topic 

effort
Senior Member
USA
38 Posts 
Posted  05/03/2013 : 10:49:44

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? 


someguy
Advanced Member
Canada
143 Posts 
Posted  05/03/2013 : 12:11:14

Hi Effort,
You are looking for tan(/2) where =arccos(5/13).
Using the halfangle identities for the tangent function is useful here.
tan(/2) = sin()/(1+cos()) = (1cos())/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).



the_hill1962
Advanced Member
USA
1469 Posts 
Posted  05/03/2013 : 18:00:36

Quick response, someguy! I was looking up the halfangle 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!



effort
Senior Member
USA
38 Posts 
Posted  05/06/2013 : 08:16:47

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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