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C3 half angle of TAN proof!

If Sin1/2x= +/- (square root) ((1-cosx)/2)
and Cos1/2x= +/- (square root) ((1+cosx)/2)

Then how does tan1/2x become sinx/(1+cosx)?

Could someone explain the steps of the proof please. Thanks
Original post by Johnpeters
If Sin1/2x= +/- (square root) ((1-cosx)/2)
and Cos1/2x= +/- (square root) ((1+cosx)/2)

Then how does tan1/2x become sinx/(1+cosx)?

Could someone explain the steps of the proof please. Thanks

sin(12x)=±1cos(x)2,cos(12x)=±1+cos(x)2\sin(\frac{1}{2}x) = \pm \sqrt{\frac{1-\cos(x)}{2}}, \cos(\frac{1}{2}x) = \pm \sqrt{\frac{1+\cos(x)}{2}}.

Then tan(12x)=1cos(x)1+cos(x)\tan(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}. Multiply top and bottom by 1+cos(x)1+\cos(x).
Reply 2
Original post by Smaug123
sin(12x)=±1cos(x)2,cos(12x)=±1+cos(x)2\sin(\frac{1}{2}x) = \pm \sqrt{\frac{1-\cos(x)}{2}}, \cos(\frac{1}{2}x) = \pm \sqrt{\frac{1+\cos(x)}{2}}.

Then tan(12x)=1cos(x)1+cos(x)\tan(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}. Multiply top and bottom by 1+cos(x)1+\cos(x).


Thanks a lot :smile:
EDIT: Don't you multiply the top and bottom by (squareroot) 1+cos(x)?
(edited 9 years ago)
Original post by Smaug123
sin(12x)=±1cos(x)2,cos(12x)=±1+cos(x)2\sin(\frac{1}{2}x) = \pm \sqrt{\frac{1-\cos(x)}{2}}, \cos(\frac{1}{2}x) = \pm \sqrt{\frac{1+\cos(x)}{2}}.

Then tan(12x)=1cos(x)1+cos(x)\tan(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}. Multiply top and bottom by 1+cos(x)1+\cos(x).


good explanation.

just wondering why the "plus or minus" is not on the tan ?
Original post by Johnpeters
Thanks a lot :smile:
EDIT: Don't you multiply the top and bottom by (squareroot) 1+cos(x)?

Yes, sorry - I meant "multiply the fraction which is inside the square root", not "multiply the whole expression". Sorry.
Original post by the bear
good explanation.

just wondering why the "plus or minus" is not on the tan ?

Consider when tan(x/2) is positive - it's exactly when x is between 0 and pi, or 2pi and 3pi, or…
Consider when sin(x) is positive - it's exactly when x is between 0 and pi, or 2pi and 3pi, or…
And 1+cos(x) is always positive.
Original post by Smaug123
Consider when tan(x/2) is positive - it's exactly when x is between 0 and pi, or 2pi and 3pi, or…
Consider when sin(x) is positive - it's exactly when x is between 0 and pi, or 2pi and 3pi, or…
And 1+cos(x) is always positive.


thanks for that :wink:

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