# Thread: General solution

1. ## General solution

Write general solutions for these trig equations

sin3a+sina = 0

cos4x+cos2x = 0

any help greatly appreciated. thanx !!

2. Hello, THSKluv!

We need two Sum-to-Product Identities:

. . $\displaystyle \begin{array}{cccc}{\color{blue}(1)} &\sin A + \sin B &=& 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \\ \\ {\color{blue}(2)} & \cos A + \cos B &=& 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \end{array}$

Write general solutions for these trig equations:

. . $\displaystyle \sin3a+\sin a \:=\: 0$
Using [1], we have: .$\displaystyle 2\sin(2a)\cos(a) \:=\:0$

Then: .$\displaystyle \begin{array}{ccccc}\sin(2a) \:=\:0 & \Rightarrow& 2a\:=\:\pi n &\Rightarrow& \boxed{a \:=\:\frac{\pi}{2}n} \\ \cos(a) \:=\:0 &\Rightarrow& \boxed{a \:=\:\frac{\pi}{2} + \pi n} \end{array}$

$\displaystyle \cos4x+\cos2x \:=\: 0$
Using [2], we have: .$\displaystyle 2\cos(3x)\cos(x) \:=\:0$

Then: .$\displaystyle \begin{array}{ccccc}\cos(3x)\:=\:0 &\Rightarrow& 3x \:=\:\frac{\pi}{2} + \pi n &\Rightarrow& \boxed{x \:=\:\frac{\pi}{6} + \frac{\pi}{3}n} \\ \cos(x) \:=\:0 &\Rightarrow& \boxed{x \:=\:\frac{\pi}{2} + \pi n} \end{array}$