Correcting the Sine Small-Angle Approximation

Calculus Level 2

What is the lowest-order correction term to the small-angle approximation for the sine function

$\sin \theta\approx \theta?$

-\frac{\theta^3}{3}

-\frac{\theta^3}{6}

\frac{\theta^3}{6}

\frac{\theta^2}{2}

1 solution

Matt DeCross
Mar 13, 2016

The small-angle approximation for the sine function comes from its Taylor series,

$\sin x = x - \frac{x^3}{6} + \mathcal{O} (x^5).$

The answer is the first correction term after $x$ , with $x = \theta$ in the problem statement.

what if sin x= x?

SfGenzkie Nga Pala - 5 years, 3 months ago

The problem does indeed take the lowest-order small-angle approximation for sin to be $\sin x\approx x$ . However, note that it is asking for the lowest-order correction to this approximation. This corresponds to the next term in the Taylor series given above.

Matt DeCross - 5 years, 2 months ago

Correcting the Sine Small-Angle Approximation

1 solution

0 pending reports