Proof of the double-angle and half-angle formulas Double-angle formulas Proof 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 To derive the third version, in line (1) use this Pythagorean identity: cos2 = 1 − sin2. We have
These are the three forms of cos 2. Half−angle formulas . . . . . . . (2') . . . . . . . (3') Whether we call the variable θ or does not matter. What matters is the form. Proof 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. Notice that this formula is labeled (2') -- "2-prime"; this is to remind us that we derived it from formula (2).
transposing, line (3) becomes
This is the half−angle formula for the sine. Table of Contents | Home Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |