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

Reply 1

Smaug123

15

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(\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(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}

. Multiply top and bottom by

1+\cos(x)

.

Reply 2

Johnpeters

OP

4

Original post by Smaug123

\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(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}

. Multiply top and bottom by

1+\cos(x)

.

Thanks a lot :smile:

EDIT: Don't you multiply the top and bottom by (squareroot) 1+cos(x)?

(edited 10 years ago)

Reply 3

the bear

20

Original post by Smaug123

\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(\frac{1}{2}x) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}

. Multiply top and bottom by

1+\cos(x)

.

good explanation.

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

Reply 4

Smaug123

15

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.

Reply 5

Smaug123

15

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.

Reply 6

the bear

20

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:

C3 half angle of TAN proof!

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