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The maximum value of $\theta$ until which the approximation $\sin\theta \approx \theta$ holds to within $10\%$ error is

  1. $10^{\circ}$
  2. $18^{\circ}$
  3. $50^{\circ}$
  4. $90^{\circ}$
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We know that,  $\displaystyle{\lim_{\theta \rightarrow 0}}\sin \theta \approx 0$

When $\theta = 10^\circ = 10^\circ \times \dfrac{\pi}{180^\circ } = 0.1745$

$\sin\theta = \sin 10^\circ = 0.1736$

$∴\% error = \dfrac{0.1745-0.1736}{0.1736}\times 100\% = 0.51\%$

When $\theta = 18^\circ = 18^\circ \times \dfrac{\pi}{180^\circ } = 0.314$

$\sin\theta = \sin 18^\circ = 0.3090$

$∴\% error = \dfrac{0.314-0.3090}{0.3090}\times 100\% = 1.6\%$

When $\theta = 50^\circ = 50^\circ \times \dfrac{\pi}{180^\circ } = 0.872$

$\sin\theta = \sin 50^\circ = 0.766$

$∴\% error = \dfrac{0.872-0.766}{0.766}\times 100\% = 13.83\%$

When $\theta = 90^\circ = 90^\circ \times \dfrac{\pi}{180^\circ } = 1.57$

$\sin\theta = \sin 90^\circ = 1$

$∴\% error = \dfrac{1.57-1}{1}\times 100\% = 57\%$

$\color{green}{\text{∴Maximum value of }}$$\color{green}{\theta}$$\color{green}{\text{ will be}}$ $\color{orange}{18^\circ}$ $\color{green}{\text{until which the approximation}}$ $\color{blue}{\sin \theta \approx \theta}$ $\color{green}{\text{holds to within}}$ $\color{blue}{10\%}$ $\color{green}{\text{error}}$.
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