\(
sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -...
\)
\( cos(x) = 1- \frac{x^2}{2!} + \frac{x^4}{4!}-... \)
here x is measured in radians. Using these formulae we can estimate the values of sine and cosine function.
For example, \(sin(380°) = sin(360°+20°) = sin(20°) = sin(20\frac{\pi}{180}) \)
substituting \(x\) by \(\frac{\pi}{9}\) we get \[ sin(\frac{\pi}{9}) \approx \frac{\pi}{9} - \frac{\pi^3}{9^33!} \approx 0.342 \] this matches approximately with the real result.
Let \(cos x = t\) and \(x^2 = X\)
\(\implies t \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} = 1 - \frac{X}{2!} + \frac{X^2}{4!} \)
\(\implies X^2 -12X+24(1-t) = 0 \)
\(\implies X = 6 \pm \sqrt{36 - 24(1-t)} \)
or, \(x = \sqrt{6-\sqrt{12+24t}} \)
or, \(\boxed{\theta \approx \sqrt{6-\sqrt{12+24cos(\theta)}}} \)
here \(\theta\) is measured in radians.
For example, \(cos(\theta) = \frac{4}{5} \implies \theta \approx 0.644^c = 36.879^\circ \)
Note that these formulae should be applied for \(0^c =0^\circ < \theta < 45^\circ = 0.7854^c \), or \( 0 < sin(\theta)< 0.7 \) , \( 0.7 < cos(\theta) < 1 \)
\( cos(x) = 1- \frac{x^2}{2!} + \frac{x^4}{4!}-... \)
here x is measured in radians. Using these formulae we can estimate the values of sine and cosine function.
For example, \(sin(380°) = sin(360°+20°) = sin(20°) = sin(20\frac{\pi}{180}) \)
substituting \(x\) by \(\frac{\pi}{9}\) we get \[ sin(\frac{\pi}{9}) \approx \frac{\pi}{9} - \frac{\pi^3}{9^33!} \approx 0.342 \] this matches approximately with the real result.
Let \(cos x = t\) and \(x^2 = X\)
\(\implies t \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} = 1 - \frac{X}{2!} + \frac{X^2}{4!} \)
\(\implies X^2 -12X+24(1-t) = 0 \)
\(\implies X = 6 \pm \sqrt{36 - 24(1-t)} \)
or, \(x = \sqrt{6-\sqrt{12+24t}} \)
or, \(\boxed{\theta \approx \sqrt{6-\sqrt{12+24cos(\theta)}}} \)
here \(\theta\) is measured in radians.
For example, \(cos(\theta) = \frac{4}{5} \implies \theta \approx 0.644^c = 36.879^\circ \)
Note that these formulae should be applied for \(0^c =0^\circ < \theta < 45^\circ = 0.7854^c \), or \( 0 < sin(\theta)< 0.7 \) , \( 0.7 < cos(\theta) < 1 \)
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