Linear approximations are used to estimate functions using derivatives
The approximated value of sin(56 degrees) is 0.8429
The trigonometry expression is given as:
[tex]\sin(56^o)[/tex]
Convert 56 degrees to radians
[tex]56^o = \frac{56}{180}\pi[/tex]
To approximate, we make use of 45 degrees.
Where:
[tex]\sin(45^o) = \cos(45^o) = \frac{\sqrt 2}{2}[/tex]
Also, we have:
[tex]45^o= \frac{\pi}{4}[/tex]
And
[tex](\sin\ x)'= \cos\ x[/tex]
So, the approximation of sin(56 degrees) become:
[tex]\sin(56\°) = \sin(45\°) + (\frac{56}{180}\pi - \frac{\pi}{4}) *\cos(45\°)[/tex]
Substitute known values
[tex]\sin(56\°) = \frac{\sqrt 2}{2} + (\frac{56}{180}\pi - \frac{\pi}{4}) *\frac{\sqrt 2}{2}[/tex]
Take LCM
[tex]\sin(56\°) = \frac{\sqrt 2}{2} + \frac{56 - 45}{180}\pi *\frac{\sqrt 2}{2}[/tex]
[tex]\sin(56\°) = \frac{\sqrt 2}{2} + \frac{11}{180}\pi *\frac{\sqrt 2}{2}[/tex]
Solve the expression
[tex]\sin(56^o) = 0.8429[/tex]
Hence, the approximated value of sin(56 degrees) is 0.8429
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