The number of integral values of k for which the equation 3 x+4 x = k+1 has a solution, kR…

Question

The number of integral values of $k$ for which the equation $3\sin x+4\cos x = k+1$ has a solution, $k\in\mathbb{R}$, is \_\_\_\_\_\_\_\_\_\_\_

Answer: 11

Accepted Answer

Writing $3\sin x+4\cos x = 5\sin(x+\alpha)$ where $\cos\alpha=\tfrac35,\sin\alpha=\tfrac45$:

$$5\sin(x+\alpha) = k+1 \implies -5\le k+1\le5 \implies -6\le k\le4$$

The integers from $-6$ to $4$ number $\boxed{11}$.

Solutions · 1