## anonymous 5 years ago how can we calculate cos. 73 without a calculator?

1. amistre64

if you know that taylor polynomial for it ....

2. mathmate

without a calculator?? :)

3. amistre64

$cos(x)=cos(0)-sin(0)x-\frac{cos(0)}{2!}x^2+\frac{sin(0)}{3!}x^3+\frac{cos(0)}{4!}x^4+...$

4. amistre64

the odd powers go to zero, and the even powers go to 1/n! x^n alternating + and -

5. anonymous

without a cal, yup :)

6. amistre64

to clean it up: $cos(x)=1-\frac{1}{2}x^2+\frac{1}{24}x^4-\frac{1}{720}x^6+\frac{1}{8!}x^8-\frac{1}{10!}x^{10}+...$

7. amistre64

and I believe the 73 needs to be turned into radians by multiplying it by pi/180

8. amistre64

i wonder if the newton method would be adaptable?

9. mathmate

Pen and paper allowed? or not even?

10. anonymous

pen and paper allowed:) just no calculator

11. mathmate

Abacus allowed?

12. anonymous

yea :D

13. amistre64

cos(73)=sin(17) is thats more doable

14. mathmate

So use the Taylor's series above. You can get answers to as accurately as the abacus allows. With pen and paper, you can get to quite a good degree of accuracy. You can also use the sum/difference formulas and double and half angle formulas to get approximations, and correct the difference using linearization. The question would be more complete if it specifies how many digits of accuracy are required. The answer would be different if 2 decimal digits are required (in that case, may be even mental calculation), 4 (pen and paper), 8 digits would require an abacus.

15. mathmate

You'll need to know how to calculate square-roots with pen and paper or by approximations, and know that pi=3.1415926535897932... cos(75)=(sqrt(6)-sqrt(4))/4 sin(75)=(sqrt(6)+sqrt(4))/4 By linearization: dcos(x)/dx=-sin(x) cos(x+d)=cos(x)+ (-sin(x))*d where d=-2 degrees in radians cos(73) =cos(75-2) =cos(75)-sin(75)*(-2pi/180) =(sqrt(6)-sqrt(4))/4 - (sqrt(6)+sqrt(4))/4 *(-2*pi/180) =0.29254 Exact value = 0.29237

16. mathmate

sorry, corrections: cos(75)=(sqrt(6)-sqrt(2))/4 sin(75)=(sqrt(6)+sqrt(2))/4 ... =(sqrt(6)-sqrt(2))/4 - (sqrt(6)+sqrt(2))/4 *(-2*pi/180)