Sum of sine and cosine

Note that $\sin x + \cos x = \sqrt 2 \sin \left(x + \frac \pi 4\right) \le \sqrt 2$ , since $-1\le \sin \theta \le 1$ . Therefore, the probability is $\boxed{1}$ .

By the auxiliary angle method, $\sin(x)+\cos(x)=\sqrt2\sin\left(x+\frac{\pi}{4}\right)$ This is found by letting $\sin x+\cos x=R\sin(x+\alpha)$ then using the compound angle rule for $\sin(A+B)$ and equating like terms to find $R$ and $\alpha$ .

The function $\sin(x)$ has a range between $1$ and $-1$ , so the largest value of $\sqrt2\sin\left(x+\frac{\pi}{4}\right)$ is $\sqrt2$ . Therefore $\sin(x)+\cos(x)\le\sqrt2$ for any real $x$ value, and so the probability must be $\boxed{1}$ .

The maximum and minimum values of $a\sin(x)+b\cos(x)$ are $+\sqrt{a^2+b^2}$ and $-\sqrt{a^2+b^2}$ .

Hence maximum value is $=\sqrt{1^2+1^2}=\sqrt{2}$ , and hence it is always true , therefore the probability will be $\boxed{1}$ ..