Trigonometry

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 = alpha.  We have

sin 2alpha = sin (alpha + alpha) = sin alpha cos alpha + cos alpha sin alpha
 
  = 2 sin alpha cos alpha.
 
cos 2alpha = cos (alpha + alpha) = cos alpha cos alpha − sin alpha sin alpha
 
cos 2alpha = cos2alpha − sin2alpha.   .  .  .  .  .  . (1)

This is the first of the three versions of cos 2alpha.  To derive the second version, in line (1) use this Pythagorean identity:

sin2alpha = 1 − cos2alpha.

Line (1) then becomes

cos 2alpha = cos2alpha − (1 − cos2alpha)
 
  = cos2alpha − 1 + cos2alpha.
 
cos 2alpha = 2 cos2alpha − 1.  .  .  .  .  .  .  .  .  .  (2)

To derive the third version, in line (1) use this Pythagorean identity:

cos2alpha = 1 − sin2alpha.

We have

cos 2alpha = 1 − sin2alpha − sin2alpha;.
 
cos 2alpha = 1 − 2 sin2alpha.  .  .  .  .  .  .  .  .  .  (3)

These are the three forms of cos 2alpha.

Half−angle formulas

 .  .  .  .  .  .  .  (2')

 .  .  .  .  .  .  .  (3')

Whether we call the variable θ or alpha does not matter.  What matters is the form.  

Proof

Now, alpha is half of 2alpha.  Therefore, in line (2), we will put 2alpha = θ, so that

   alpha becomes  θ
2
:
cos θ = 2 cos2 θ
2
 − 1.
On solving this algebraically for cos  θ
2
, we will have the half-angle

formula for the cosine.

So, on transposing 1 and exchanging sides, we have

2 cos2 θ
2
= 1 + cos θ
 
cos2 θ
2
= ½(1 + cos θ)
 
cos  θ
2
= .

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 alpha does not matter.

Notice that this formula is labeled (2') -- "2-prime"; this is to remind us that we derived it from formula (2).

The formula for sin  θ
2
 comes from putting 2alpha = θ in line (3).  On

transposing, line (3) becomes

  2 sin2 θ
2
= 1 − cos θ,
  so that
 
  sin  θ
2
= .

This is the half−angle formula for the sine.


Trigonometric identities


Table of Contents | Home


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2015 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com