Calculate Sin/Cos/Tan By Hand - No Calculator Needed!

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SUMMARY

This discussion focuses on calculating sine values by hand for angles close to 42°, 43°, and 44° using numerical methods. The method involves converting degrees to radians and applying the sine addition formula: \(\sin(\pi/4 + x) = \sin(\pi/4)\cos(x) + \sin(x)\cos(\pi/4)\). The values of \(x\) are derived from small increments in radians, specifically \(-\pi/180, -2\pi/180, -3\pi/180\). By utilizing the Taylor series for sine and cosine near zero, one can achieve accurate approximations for these sine values.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with radians and degree conversion
  • Knowledge of the Taylor series expansion for sine and cosine
  • Ability to perform basic algebraic manipulations
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  • Study the sine addition formula in detail
  • Learn about Taylor series and their applications in trigonometry
  • Explore numerical methods for approximating trigonometric functions
  • Practice converting between degrees and radians for various angles
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Students of mathematics, educators teaching trigonometry, and anyone interested in manual calculations of trigonometric functions without the use of calculators.

Goliatbagge
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Does it exist a numerical method to calculate for example sin (42°), sin (43°), sin (44°) etc by hand?
 
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Goliatbagge said:
Does it exist a numerical method to calculate for example sin (42°), sin (43°), sin (44°) etc by hand?

Of course you will turn these into radians ... but since all of your values are near \pi/4, I would use,
<br /> \sin ( \pi/4 + x) = \sin(\pi/4) \cos(x) + \sin(x) \cos(\pi/4) = \left(\cos(x) + \sin(x) \right) / \sqrt{2}.<br />
Your values of x are -\pi/180, -2 \pi/180, - 3 \pi/180, ... which are small, so you can take the first few terms in the Taylor series for sin and cos near zero. Even approximating \cos(x) \approx 1 and \sin(x)\approx x should get you several correct digits for the exact examples you gave.

jason

EDIT: I now see that this was included in adjacent's link above. I had just read the first part with the Taylor series about zero ... Oops!
 
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