Linear equations in sine and cosine

The parametric formulae are the key to solving linear equations in $\sin$ and $\cos$, that is, equations of the form $a\sin\alpha + b\cos\alpha + c = 0$.

Observation — When they are needed

By setting $t=\tan(\alpha/2)$ one obtains $a\cdot\frac{2t}{1+t^2} + b\cdot\frac{1-t^2}{1+t^2} + c = 0,$ which, on multiplying by $1+t^2$, becomes a rational equation in $t$, in general of the second degree: the trigonometric problem is thus transformed into pure algebra. The explicit method will be taken up again in the chapter on trigonometric equations.

Warning — The "lost" solution

In passing from $\alpha$ to $t=\tan(\alpha/2)$ one implicitly loses the case $\alpha = \pi + 2k\pi$, where $\tan(\alpha/2)$ does not exist. This solution must always be checked separately, by substituting $\alpha=\pi$ into the original equation.

Links

Topics: Prostaferesi werner
Concepts: Equazioni lineari in seno e coseno · Formule parametriche · Tangente dell angolo meta
Functions: Coseno · Seno
Skills: Risolvere equazioni · Usare formule