5.5 | Multiple-Angle and Product-to-Sum Formulas


Double-Angle Formulas

  • \(\displaystyle \sin 2u = 2 \sin u \cos u\)
  • \(\displaystyle \begin{aligned} \cos 2u & = \cos^2 u – \sin^2 u \\
    & = 2 \cos^2 u -1 \\
    & = 1-2\sin^2 u
    \end{aligned}
    \)
  • \(\displaystyle \tan 2u = \frac{2 \tan u}{1-\tan^2 u}\)
Power-Reducing Formulas

  • \(\displaystyle \sin^2u = \frac{1-\cos 2u}{2}\)
  • \(\displaystyle \cos^2u = \frac{1+\cos 2u}{2}\)
  • \(\displaystyle \tan^2 u = \frac{1-\cos 2u}{1+ \cos 2u}\)
Sum-to-Product Formulas

  • \(\displaystyle \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \sin u – \sin v = 2 \cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u + \cos v = 2 \cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u – \cos v = -2 \sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
Half-Angle Formulas

  • \(\displaystyle \sin \frac{u}{2} = \pm \sqrt{\frac{1-\cos u}{2}}\)
  • \(\displaystyle \cos \frac{u}{2} = \pm \sqrt{\frac{1+\cos u}{2}}\)
  • \(\displaystyle \tan \frac{u}{2} = \frac{1-\cos u}{\sin u} = \frac{\sin u}{1+\cos u}\)

The signs of \(\sin\frac{u}{2}\) and \(\cos \frac{u}{2}\) depend on the quadrant in which \(\frac{u}{2}\) lies.

Product-to-Sum Formulas

  • \(\displaystyle \sin u \sin v = \frac{1}{2}[\cos(u-v) – \cos(u+v)]\)
  • \(\displaystyle \cos u \cos v = \frac{1}{2}[\cos(u-v) + \cos(u+v)]\)
  • \(\displaystyle \sin u \cos v = \frac{1}{2}[\sin(u+v) + \sin(u-v)]\)
  • \(\displaystyle \cos u \sin v = \frac{1}{2}[\sin(u+v) – \sin(u-v)]\)
Sum-to-Product Formulas

  • \(\displaystyle \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \sin u – \sin v = 2 \cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u + \cos v = 2 \cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u – \cos v = -2 \sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)

E 5.5 Exercises