Question

how can we calculate cos. 73 without a calculator?

Answer 1

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

Answer 2

without a calculator?? :)

Answer 3

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

Answer 4

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

Answer 5

without a cal, yup
:)

Answer 6

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

Answer 7

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

Answer 8

i wonder if the newton method would be adaptable?

Answer 9

Pen and paper allowed? or not even?

Answer 10

pen and paper allowed:) just no calculator

Answer 11

Abacus allowed?

Answer 12

yea
:D

Answer 13

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

Answer 14

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.

Answer 15

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

Answer 16

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)