SOLUTION: tan(1/2cos^-1(6/7))

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Question 1079196: tan(1/2cos^-1(6/7))
Answer by ikleyn(52906) About Me  (Show Source):
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tan(1/2cos^-1(6/7))
~~~~~~~~~~~~~~~ 

We need to find tan(1/2cos^-1(6/7)), which is tan%28%281%2F2%29%2Aarccos%286%2F7%29%29.

Let alpha = arccos%286%2F7%29.


It means that alpha is the angle in QI such that cos%28alpha%29 = 6%2F7.

And we need to find tan%28alpha%2F2%29 = sin%28alpha%2F2%29%2Fcos%28alpha%2F2%29.


According to the formulas of half-angle 

    (they are basic formulas of Trigonometry; see the lesson Trigonometric functions of half argument in this site)


sin%28alpha%2F2%29 = sqrt%28%281-cos%28alpha%29%29%2F2%29 = sqrt%28%281-6%2F7%29%29%2F2 = sqrt%28%287-6%29%2F%287%2A2%29%29 = sqrt%281%2F14%29,


cos%28alpha%2F2%29 = sqrt%28%281%2Bcos%28alpha%29%29%2F2%29 = sqrt%28%281%2B6%2F7%29%29%2F2 = sqrt%28%287%2B6%29%2F%287%2A2%29%29 = sqrt%2813%2F14%29.


Then  tan%28alpha%2F2%29 = sin%28alpha%2F2%29%2Fcos%28alpha%2F2%29 = sqrt%281%2F14%29%2Fsqrt%2813%2F14%29 = sqrt%281%2F13%29.

Answer. tan(1/2cos^-1(6/7)) = tan%28%281%2F2%29%2Aarccos%286%2F7%29%29 = sqrt%281%2F13%29.

Solved.


If you want to see more solved problems of this kind, look into the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Trigonometry: Solved problems".