How do you solve cos 100 without a calculator?

Answer 1

(Assuming 100 is 100 degrees)

cos 100 degrees is equal to - sin 10 degrees.

In radians, this is - sin (pi/18).

Approximating pi as 22/7, this is - sin (11/63)

Using four terms of a Taylor series, this is approximately:
- (11/63) + (11/63)^3 /6 - (11/63)^5 / 120 + (11/63)^7 / 5040.

- (11/63) + (11^3/63^3) /6 - (11^5/63^5) / 120 + (11^7/63^7) / 5040.

Rewriting the fractions with the common denominator 63^7:
- (11*63^6/63^7) + (11^3*63^4/63^7) /6 - (11^5*63^2/63^7) / 120 + (11^7/63^7) / 5040.

Rewriting again with common denominators of 5040:
- 5040*(11*63^6/63^7)/5040 + 840*(11^3*63^4/63^7)/5040 - 42*(11^5*63^2/63^7) / 5040 + (11^7/63^7) / 5040.

Now add it all up:
[- 5040*11*63^6 + 840*11^3*63^4 - 42*11^5*63^2 + 11^7] / (5040*63^7)

Now, some seriously long multiplication gives:
(- 3466302962466960 + 17612440516440 - 26846879598 + 19487171) / 19852462421401680

Some really easy addition:
-3448717349342947 / 19852462421401680

And finally, do the long division:
-0.17371735939 ... and so on.

The actual value is -0.17364817766693034885171662676931

So, I got 3 decimal places right. Maybe use a better approximation of pi (like 355/113) and more terms of the Taylor series. But you get the picture.