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Surd form of tan(3p/8)
By Anonymous on Sunday, April 15, 2001 - 01:40 pm:

Hi,

How do you get the surd result of tan(3p/8)?

Thanks for your help in advance.

By Anonymous on Sunday, April 15, 2001 - 02:11 pm:

Well, you know the tan of 3p/4, which is -1. This is equal to 2t/(1-t2), where t is the tan of 3p/8. Simply equate the two values and find t using the quadratic formula.

By David Loeffler (P865) on Sunday, April 15, 2001 - 09:33 pm:

This method will of course return two answers, and you will have to check their signs to see which one is positive. (One is tan(3p/8), the other is tan(7p/8), since tan(7p/4) = -1 as well.)

By David Loeffler (P865) on Sunday, April 15, 2001 - 09:40 pm:

PS. There is an alternative method which relies on 'pure' geometry, using some results about angle bisectors of triangles. This avoids solving any quadratics, which is a bit more elegant. Let me know if you're interested.

By Olof Sisask (P3033) on Monday, April 16, 2001 - 05:06 pm:

I'd be interested to hear your method David.

/Olof.

By Kerwin Hui (Kwkh2) on Monday, April 16, 2001 - 05:52 pm:

Let me guess David's method:

Construct triangle ABC with AB=AC and angle BAC=p/2. Draw the angle bisector of angle ABC to meet BC at P. Now, by the angle bisector theorem,

AP/AB=PC/BC

but PC=AC-PA and BC=sqrt(2)×AB

so sqrt(2)×AP=AC-AP

rearrange for AC/AP, which is tan(3p/8).

Kerwin