[chongli]

You can teach the students trig identities to expand the choice of angles. Moreover, the purpose of the lesson should be to teach the method of solving the problem. Pushing the buttons on the calculator to get a final answer in decimal form (only an approximation) is extraneous to that goal.

[mbreese]

Or they could go really old-school and provide a look up table...

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.

[rtkwe]

It's been a long time since I had to use the identities but looking over them quickly I don't see how they help you get the approximation of cos(70 deg) knowing the 5 basic values that I memorized back in school of (0,30,45,60,90).

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.

[chongli]

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.

[lasagnaphil]

You can get close-enough ballpark answers on the angles between (0, 30, 45, 60, 90). (If the questions are multiple choice, then you can design it in a way that you don't need a calculator) And this is actually quite an important handy skill you will have in real life, since later on when you're drawing graphics/diagrams either on software or hardware you really need that intuition of ball-parking angles.