Finding values of sine without a calculator

How do I find the sin or inverse sin of 46 degrees (or any degree) without using a calculator?

Finding an "inverse sin" doesn't apply to degrees by the way. We take sine of an angle and get a number; inverse sine is taken from a number and gives back an angle.

So is it possible to find values of sine without a calculator? I am sure there are various methods, but these two came to my mind.

1) We can go back to the definition of sine in a right triangle and using a protractor, DRAW a right triangle with that angle. Draw as accurately as you can.

Then from the picture again, we need to measure the two sides: the opposite side and the hypotenuse and then calculate their ratio (paper and pencil). Again, measure as accurately as you can.

Now, as far as the opposite problem, let's say you know that the sine of some angle is 0.86 or some other number (between -1 and 1). Can we find the angle without the calculator?

Draw a right triangle with hypotenuse 1, opposite side 0.86 (or some multiple of those), and measure the angle in degrees.

This method, I feel, is a good one for demonstration and teaching purposes when first learning about sine.

2) Another possibility is to use the Taylor series of sine. Hopefully you don't need to take too many terms from it to get the desirable accuracy. This will include calculations that you'd need to do on paper.

Let's say we use Taylor series in origin and take the first four terms:

sin(x) ≈ x - x3/3! + x5/5! - x7/7!

To use this, you need to first change the 46 degrees or whatever to radians. Obviously all the calculations involved will take some time without a calculator... Auch! But an approximative method such as this that only involves the four basic operations is what your calculator probably uses, too.

Here you will find the Taylor series for inverse of sine.

3) Using sine addition formula and a known value.

Sine addition formula says:
sin(a + b) = sin a cos b + cos a sin b.

So... if you're interested in finding, say, sin(46°) and we do already happen to know sin(45°) and cos(45°)... but we need to work in radians to use the formula. So convert 46 and 45 degrees to radians (without a calculator? I'm going to cheat now...) and get 45° is Pi/4 or 0.785398, 46° is about 0.80285.

sin(Pi/4 + 0.01745) = sin(Pi/4) cos(0.01745) + cos(Pi/4) sin(0.01745).

It just so happens that for small values of x (near zero), sin x ≈ x. (You learn that in calculus, I think). So sin(0.01745) is about 0.01745. Cos(0.01745) should be pretty near 1 somewhere.

sin(Pi/4) or sin 45° is 1/√2, and cos(Pi/4) is the same. Plugging those in,

sin(Pi/4 + 0.01745) = 1/√2 * 1 + 1/√2 *0.01745.

I'm going to cheat again and do this with a calculator... to get 0.71945.

Did I get close? Well, sin46° is about 0.7193398. Got three decimals right; that's okay I guess.

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