If $\cos(A)-4\sin(A)=1$ then what are the possible values of $\sin(A)+4\cos(A)$?

Question

Given that

$$\cos(A)-4\sin(A)=1.$$

What are the possible values of $\sin(A)+4\cos(A)$?

This is rather a very difficult and unfamiliar question, any help is much appreciated.

Answer 1

$$(\cos A-4\sin A)^2+(4\cos A+\sin A)^2=?$$

This is a corollary of Brahmagupta–Fibonacci identity

Answer 2

$\displaystyle \cos A - 4\sin A = \sqrt{17}\cos(A + \arctan 4)$

$\displaystyle \sin A + 4\cos A = \sqrt{17}\sin (A + \arctan 4)$

Hence the latter is equal to $\displaystyle \pm\sqrt{17}\sqrt{1 - (\frac{1}{\sqrt{17}})^2} = \pm 4$

Answer 3

$$\cos(A)−4\sin(A)=1 \iff \cos(A)=1+4\sin(A)$$ Substitute this $\cos(A)$ in $\sin(A)+4\cos(A)$:

$$\begin{align*}\sin(A)+4\cos(A)&=\sin(A)+4(1+4\sin(A))\\ &=\sin(A)+4+16\sin(A)\\ &=4+17\sin(A)\end{align*}$$

The range of $\sin(A)$ is $[-1,1]$, so the range of $\sin(A)+4\cos(A)$ will be $[-13,21]$.