Question

The absolute error in the value of sin𝜃 if approximated up to two terms in the Taylor’s series for 𝜃 = 60∘ is ___________ (rounded off to three decimal places).

Answer 1

Correct Answer: 0.009 - 0.011

Given: 
Angle \( \theta = 60^\circ = \frac{\pi}{3} = 1.0472\ \text{rad} \)

Taylor series of sin θ:
\[ \sin\theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \cdots \] Using only first two terms: \[ \sin\theta \approx \theta - \frac{\theta^3}{6} \] Step 1: Compute approximation
\[ \theta = 1.0472 \] \[ \theta^3 = 1.0472^3 \approx 1.148 \] \[ \frac{\theta^3}{6} = \frac{1.148}{6} \approx 0.1913 \] \[ \sin\theta_{\text{approx}} = 1.0472 - 0.1913 = 0.8559 \] Step 2: True value \[ \sin 60^\circ = 0.8660 \] Step 3: Absolute error
\[ |\text{error}| = |0.8660 - 0.8559| = 0.0101 \] Final Answer:
\[ \boxed{0.010} \]