1. trigo problem no.6

trigo problem no.6
p181 q11 ex8b
question:
show that the equation acosx + bsinx +c = 0 has only one root in the range of 0 =< x < 360 if and only if a^2 + b^2 + c^2 .

though i know the discrimination D = 0 i don't know how to do the proving as it is not a quadratic equation.

2. Originally Posted by afeasfaerw23231233
trigo problem no.6
p181 q11 ex8b
question:
show that the equation acosx + bsinx +c = 0 has only one root in the range of 0 =< x < 360 if and only if a^2 + b^2 = c^2 . Mr F edit correction red.

though i know the discrimination D = 0 i don't know how to do the proving as it is not a quadratic equation.
You should be comfortable with the fact that $\displaystyle a \cos x + b \sin x$ can be expressed in the form $\displaystyle \sqrt{a^2 + b^2} \cos(x + \phi)$, say.

So the equation can be re-written as $\displaystyle \sqrt{a^2 + b^2} \cos (x + \phi) = -c \Rightarrow \cos (x + \phi) = - \frac{c}{\sqrt{a^2 + b^2}}$.

Now note that this equation will only have one solution over the given domain if $\displaystyle \cos(x + \phi) = \pm 1$.

So $\displaystyle -\frac{c}{\sqrt{a^2 + b^2}} = \pm 1$.

Square both sides and re-arrange: $\displaystyle c^2 = a^2 + b^2$.