sin(x) = y

for a variety of values of x.

Or, even nicer -- instead of having a correct numerical answer, they could write the answers as formulas. Then you wouldn't need to know the actual values at all. (Then again, this could also be a pain).

But some combination of the two should be possible without needing manual computation.

Even if getting the answer needs a calculator the process to get to that point is still testing the algebraic/calculus/trig/geometry skills the test is supposed to test. You can often get rid of the need for a calculator by formulating the questions carefully so the numbers are simple to work with but sometimes you can't do that easily.

You can also just accept cos(70°) as a final answer without asking students push buttons on a calculator.

For more advanced students you could teach them to calculate even weird values like 70° using a few terms of the Maclaurin series for cosine:

cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + O(x^8)

But that's all beside the point. Mathematics education should focus on understanding and applying the concepts, not using technology. The educators' rationale that "students need to learn how to use technology" doesn't really hold water when it comes to calculators. It's far easier to teach someone with a degree in mathematics how to operate a calculator, having never used one, than it is for someone totally dependent on a calculator for even basic calculations to obtain a math degree.

You /can/ but by the the time that is introduced, which is usually college since we're talking calculus, you don't want to test basic trigonometry. The tests at that level are about something else not about the particular values of the basic trig functions. Personally once I got to calculus very few of the questions were about the numerical result of a function.