### Instruction: In right triangle $ABC$ with $\angle B = 90^\circ$, we have $\sin A = 2\cos A$.  What is $\cos A$?### Response: I know that the sine and cosine of an angle in a right triangle are related to the opposite and adjacent sides of the angle, respectively.
So, if $\sin A = 2\cos A$, that means that the ratio of the opposite side to the adjacent side is 2:1.
I can use the Pythagorean theorem to find the length of the hypotenuse in terms of the adjacent side.
If I let $x$ be the adjacent side, then the opposite side is $2x$ and the hypotenuse is $\sqrt{x^2 + (2x)^2} = \sqrt{5x^2}$.
Now, I can use the definition of cosine again to write $\cos A$ as the ratio of the adjacent side to the hypotenuse.
That is, $\cos A = \frac{x}{\sqrt{5x^2}}$.
I can simplify this expression by dividing both the numerator and the denominator by $x$.
This gives $\cos A = \frac{1}{\sqrt{5}}$.
This is the exact value of $\cos A$, but I can also approximate it using a calculator if needed.
The decimal value of $\cos A$ is about 0.447.