The equation 8sinθ + 15cosθ = k has a real solution θ exactly when:

Write 8sinθ + 15cosθ = $R$·sin(θ + α) with $R$ = √$(8^2+15^2)=$ √289 = 17 and tanα = 15/8. As θ sweeps every angle, sin(θ + α) sweeps −1 to 1, so the left side reaches every value from −17 to 17 and nothing outside. Adding the coefficients as if both ratios could be 1 at once gives the impossible bound 23; 17 is only the largest value, not the only attainable one; and the expression is just as free to be negative as positive, so restricting to 0 ≤ k throws half the range away.