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anonymous

  • 5 years ago

how can we calculate cos. 73 without a calculator?

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  1. amistre64
    • 5 years ago
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    if you know that taylor polynomial for it ....

  2. mathmate
    • 5 years ago
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    without a calculator?? :)

  3. amistre64
    • 5 years ago
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    \[cos(x)=cos(0)-sin(0)x-\frac{cos(0)}{2!}x^2+\frac{sin(0)}{3!}x^3+\frac{cos(0)}{4!}x^4+...\]

  4. amistre64
    • 5 years ago
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    the odd powers go to zero, and the even powers go to 1/n! x^n alternating + and -

  5. anonymous
    • 5 years ago
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    without a cal, yup :)

  6. amistre64
    • 5 years ago
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    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
    • 5 years ago
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    and I believe the 73 needs to be turned into radians by multiplying it by pi/180

  8. amistre64
    • 5 years ago
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    i wonder if the newton method would be adaptable?

  9. mathmate
    • 5 years ago
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    Pen and paper allowed? or not even?

  10. anonymous
    • 5 years ago
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    pen and paper allowed:) just no calculator

  11. mathmate
    • 5 years ago
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    Abacus allowed?

  12. anonymous
    • 5 years ago
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    yea :D

  13. amistre64
    • 5 years ago
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    cos(73)=sin(17) is thats more doable

  14. mathmate
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    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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