From the standard expansion series, when $x$ is small and ${{x}^{3}}$ and powers above can

Question

From the standard expansion series, when $x$ is small and ${{x}^{3}}$ and powers above can be ignored (i.e.${{x}^{3}}\approx 0$). Then, the standard series from MF 26 for trigonometric ratios are reduced to $\sin x\approx x$, $\cos x\approx 1-\frac{{{x}^{2}}}{2}$. By using $\tan x=\frac{\sin x}{\cos x}\approx \frac{x}{\left( 1-\frac{{{x}^{2}}}{2} \right)}$, we can show that $\tan x\approx x$. Typical Steps for Small Angles Approximation Problems Step I To solve for length/angle geometrically • Sine Rule: $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ • Cosine Rule: ${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$ • Trigo Ratios: TOA CAH SOH Step II To separate the constant angle and variable angle as to avoid including addition of constant in the expansion. • $\sin \left( A\pm B \right)$ • $\cos \left( A\pm B \right)$ • $\tan \left( A\pm B \right)$ Step III Use standard series for trigo ratios depending on what terms are ignored. ${{x}^{3}}$ and above ignored ${{x}^{4}}$ and above ignored $\sin x\approx $ $x$ $x-\frac{{{x}^{3}}}{3!}$ $\cos x\approx $ $1-\frac{{{x}^{2}}}{2!}$ $1-\frac{{{x}^{2}}}{2!}$ Step IV Using power series, ${{\left( 1+x \right)}^{n}}$

Accepted Answer

For small angles $x$ measured in radians, $\displaystyle \sin x\approx x$, $\displaystyle \cos x\approx1-\frac{x^2}{2}$, and $\displaystyle \tan x\approx x$. Use geometry first, separate constant and variable angles, select the required truncation order, then apply any remaining power-series algebra consistently.

Small-Angle Approximations

Small-Angle Approximations

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