An airplane flies $520$ miles from Chicago to Virginia. Then it turns $45$ degrees to face New York and flies $630$ miles to New York. What is the distance from Chicago to New York?
Given the $45$ degrees to start with I've attempted to get the answer of $713$ miles, which I looked up, but when I divide $520/ cos(45)$, I get $735$ on my calculator. Is there inaccuracy in this problem, or am I doing a portion of
it incorrectly?
It can't possibly be $520/\cos(45^{\circ}),$ because that is completely independent of the second leg of the journey.
You seem to have assumed that the triangle formed by Chicago, Virginia, New York is a right triangle. That is not assumed in the question.
The correct answer (assuming the world is flat) is to use the law of cosines:
$$d^2=520^2+630^2-2\cdot 520\cdot 630\cdot\cos(135^{\circ})$$
Note: A turn of $45^{\circ}$ gives a triangle with an angle of $135^{\circ}$.
Solve for $d$. You get a much larger value this way - the triangle is obtuse, so the distance is greater than $520$ and $630$.