value of sinx + cosx + 3 greater than or less than 1

Question

Is the value of the expression: sin(x) + cos(x) + 3 always more than 1? I know how to graph it and see that this is true, but is there another way?

Answer 1

Use that $$\sin(x)+\cos(x)=\sqrt{2} \sin \left(x+\frac{\pi }{4}\right)$$

Answer 2

Sketch a rough graph of cos(x) and sin(x). (The range of) both functions are always between -1 and 1.

(The range of) both sin(x) and cos(x) are never greater than 1 nor less than -1.

So therefore, for example, (the range of) 10 + sin(x) must always be between 9 and 11.

What does this tell you about (the range of) f(x) = sin(x) + cos(x) ?