Number of integral values of $\lambda$

Question

From$\lambda cosx> -2$

how they have derived the value of $\lambda$

Answer 1

For all $x \in \mathbb R$, we have $$- 1\leq \cos x \leq 1$$ so if $\lambda \geq 0$ one has $$ - \lambda \leq \lambda\cos x \leq \lambda$$ So if one wants $ \lambda\cos x >-2$ one must have $-\lambda \geq -2$ that is to say $\lambda \leq 2$. Since $\lambda$ is an integer $\lambda = 0,1,2$.

Now if $\lambda < 0$ one has $$ - \lambda \geq \lambda\cos x \geq \lambda$$ so $\lambda \geq -2$ that implies $\lambda = -1,-2$ (using the same argument).