### Instruction: Let $a$ and $b$ be real numbers.  Find the maximum value of $a \cos \theta + b \sin \theta$ in terms of $a$ and $b.$.### Response: I notice that the expression $a \cos \theta + b \sin \theta$ looks like the dot product of two vectors, one with components $(a,b)$ and the other with components $(\cos \theta, \sin \theta).$
I recall that the dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.
So I can rewrite the expression as $|(a,b)| \cdot |(\cos \theta, \sin \theta)| \cos \phi,$ where $\phi$ is the angle between the two vectors.
I know that the magnitude of $(\cos \theta, \sin \theta)$ is always 1, since it is a unit vector on the unit circle.
So I can simplify the expression to $|(a,b)| \cos \phi.$
Now I want to maximize this expression with respect to $\phi.$
I know that the cosine function is maximized when its argument is zero, and it has a maximum value of 1.
So the maximum value of $|(a,b)| \cos \phi$ occurs when $\phi = 0,$ which means the two vectors are parallel.
Therefore, the maximum value of $a \cos \theta + b \sin \theta$ is equal to $|(a,b)|,$ which is the magnitude of the vector $(a,b).$
I can use the Pythagorean theorem to find the magnitude of $(a,b)$ in terms of $a$ and $b.$
I get $|(a,b)| = \sqrt{a^2 + b^2}.$