anonymous
  • anonymous
how can we calculate cos. 73 without a calculator?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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amistre64
  • amistre64
if you know that taylor polynomial for it ....
mathmate
  • mathmate
without a calculator?? :)
amistre64
  • amistre64
\[cos(x)=cos(0)-sin(0)x-\frac{cos(0)}{2!}x^2+\frac{sin(0)}{3!}x^3+\frac{cos(0)}{4!}x^4+...\]

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More answers

amistre64
  • amistre64
the odd powers go to zero, and the even powers go to 1/n! x^n alternating + and -
anonymous
  • anonymous
without a cal, yup :)
amistre64
  • amistre64
to clean it up: \[cos(x)=1-\frac{1}{2}x^2+\frac{1}{24}x^4-\frac{1}{720}x^6+\frac{1}{8!}x^8-\frac{1}{10!}x^{10}+...\]
amistre64
  • amistre64
and I believe the 73 needs to be turned into radians by multiplying it by pi/180
amistre64
  • amistre64
i wonder if the newton method would be adaptable?
mathmate
  • mathmate
Pen and paper allowed? or not even?
anonymous
  • anonymous
pen and paper allowed:) just no calculator
mathmate
  • mathmate
Abacus allowed?
anonymous
  • anonymous
yea :D
amistre64
  • amistre64
cos(73)=sin(17) is thats more doable
mathmate
  • 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.
mathmate
  • 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
mathmate
  • 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)

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