There are a number of problems that do not come out correctly if you use degrees instead of radians. The reason is that degrees are not a natural unit in a mathematical sense. For example, if you want to calculate the arc length of a piece of a circle that subtends an angle $\theta$, you can use the formula $S=r\theta$. If you use degrees here, you get a terrible answer. More specifically, say the radius of the circle is 2 metres and the angle subtended by the arc is $\pi/3$ radians (or 60$^\circ$), the arc length of this piece of the circle calculated using radians is $2*\pi/3$ metres which is just a bit more than 2 metres. That seems reasonable given the size of the circle. If you use degrees, you get 120 metres which is clearly ridiculous.
Any angle plugged into a trig function must be in radians but, because degrees are so common outside of a math class, calculators are designed to handle degrees inside trig functions. Formally, a calculator is actually using the function $\cos(\pi x/180)$ when you set it to accept x measured in degrees so it's doing the conversion for you.
Another place degrees cause problems is for a function like $f(t)=\sin(a(t))$. If you assume that it's ok to measure $a(t)$ in degrees, when you take the derivative of $f$ you get $f'(t)=a'(t)\cos(a(t))$. This rate of change will be in degrees per minute. If you convert to radians per minute, you end up with $\frac{\pi}{180}a'(t)\cos(a(t)$. If you instead assume that $a(t)$ is measured in radians, then $f'(t)=a'(t)\cos(a(t))$ and the value of this rate of change is in radians per minute already. These answers differ by a factor of $\pi/180$ so they are clearly not both correct. The radians answer is the correct one.