PART 5：例題-三角函數(05:28)

試以微分公式估計 \(\sin {31^ \circ }\) 之近似值

SOL:

(1)設 \(f(x) = \sin x\) ， \(f'(x) = \cos x\)

(2)微分公式 \(f(x + \Delta x) \buildrel\textstyle.\over= f(x) + f'(x)\Delta x\)

(3)三角函數中的度要換成弧度才能計算 \({31^ \circ } = {30^ \circ } + {1^ \circ } = \frac{\pi }{6} + \frac{\pi }{{180}}\)

令 \(x = \frac{\pi }{6}\) ， \(\Delta x = \frac{\pi }{{180}}\)

\(f(\frac{\pi }{6} + \frac{\pi }{{180}}) \buildrel\textstyle.\over= f(\frac{\pi }{6}) + f'(\frac{\pi }{6})\left( {\frac{\pi }{{180}}} \right)\)

\(f({31^ \circ }) \buildrel\textstyle.\over= \sin \frac{\pi }{6} + \cos \frac{\pi }{6}\left( {\frac{\pi }{{180}}} \right)\)  \( = \frac{1}{2} + \frac{{\sqrt 3 }}{2} \cdot \frac{\pi }{{180}}\)  \( \buildrel\textstyle.\over= 0.5 + 0.0{\rm{151}}\)