Proof of the double-angle and half-angle formulas
The double-angle formulas are proved from the sum formulas by putting β = . We have
This is the first of the three versions of cos 2. To derive the second version, in line (1) use this Pythagorean identity:
sin² = 1 − cos².
Line (1) then becomes
cos² = 1 − sin².
These are the three forms of cos 2.
. . . . . . . (2')
. . . . . . . (3')
Whether we call the variable θ or does not matter. What matters is the form.
Now, is half of 2. Therefore, in line (2), we will put 2 = θ, so that
formula for the cosine.
So, on transposing 1 and exchanging sides, we have
This is the half-angle formula for the cosine. The sign ± will depend on the quadrant of the half-angle. Again, whether we call the argument θ or does not matter.
transposing, line (3) becomes
This is the half−angle formula for the sine.
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Copyright © 2012 Lawrence Spector
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