Double-angle and half-angle formulas

Double-angle formulas. By letting $ v = u$ in (3-1)-(3-2), we can derive

  $\displaystyle \cos 2u = \cos^2 u - \sin^2 u = 1 - 2 \sin^2 u = 2 \cos^2 u - 1$ (3-5)
  $\displaystyle \sin 2u = 2 \sin u \cos u$    

Half-angle formulas. By letting $ u = \dfrac{v}{2}$ in (3-5), the two rightmost identities of (3-5) can be used to verify the following half-angle formulas:

    $\displaystyle \sin\dfrac{v}{2} = \pm \sqrt{\dfrac{1 - \cos v}{2}};
\quad \cos\dfrac{v}{2} = \pm \sqrt{\dfrac{1 + \cos v}{2}};$  
    $\displaystyle \tan\dfrac{v}{2} = \pm \sqrt{\dfrac{1 - \cos v}{1 + \cos v}}
= \dfrac{1 - \cos v}{\sin v}
= \dfrac{\sin v}{1 + \cos v}.$  



Department of Mathematics
Last modified: 2005-09-29