# Trig Formula Question

• StephenDoty
In summary, there are various ways to find the sin or cos of an angle without using a calculator. These include using double/triple angle identities, power series, and mental math tricks such as using exact sine ratios of angles to create linear approximations. However, the accuracy of these methods may vary and it ultimately depends on the level of accuracy needed for the specific angle.

#### StephenDoty

Is there any way to find what sin(x)= or cos(x)= when the angle is not one of the main unit circle angles without using a calculator? Like when you don't have a calculator and need to find the sin or cos of an angle.

Thank you.

Stephen

Well there's always double/triple angle identities.

You could also do a power series and add up the first few terms, I guess...

are there any mental math tricks or a way to visualize what the sin or cos of some angle equals? I mean you can't really carry around a calculator everywhere.

Any help would be appreciated.

Stephen

Hi Stephen! For small angles, sinx = x - x³/6, and cox = 1 - x²/2 + x^4/24, are accurate enough.

For example, it gives sin30º = 0.4997 and cos30º = 0.8661.

Is that close enough? StephenDoty said:
are there any mental math tricks or a way to visualize what the sin or cos of some angle equals? I mean you can't really carry around a calculator everywhere.

Any help would be appreciated.

Stephen

Double and triple identities don't require a calculator, neither does doing an infinite series if the angle is small enough...

How accurate do you wish your result to be? With basic Trig identities one can make the computation of the sin/cos of any angle into the sin/cos of an angle less than 45 degrees.

After that, either use sin or cos (30 degrees +/- x) expansions to reduce the problem to the sin/cos of an angle less than 20ish degrees, then use sin/cos (15 degrees +/- x) expansions to make the problem angle even smaller. At this point, everything else is still exact, now just approximate your small angle with the power series.

If you don't need it too exact, just a few d.p's, then Remember exact sine ratios of angles 0, 15, 30, 45, 60, 75, 90 degrees and use create a linear approximation from the one closest required angle.