The expression acos(x) + bsin(x) can be written as Rcos(x - phi) where R = sqrt($a^2$ + $b^2$) and tan(phi) = $\frac{b}{a}$. Equivalently, it equals Rsin(x + psi) where tan(psi) = $\frac{a}{b}$. Since cos and sin range from -1 to 1, the expression ranges from -R to R. For equations acos(x) + bsin(x) = c, a solution exists if and only if |c| <= R = sqrt($a^2$ + $b^2$). This method is used for: finding maximum/minimum values, solving equations that mix sin and cos linearly, and determining the range of trigonometric expressions. In JEE, this appears in optimization and equation-counting problems.