The signal response L L\,L of a specialized optical sensor is modeled by the function L(θ)=(2−12θ)6L(\theta) = \left(2 - \frac{1}{2}\theta\right)^6L(θ)=(2−21θ)6 where θ \theta\,θ represents the incident angle in radians.
Determine the first four terms, in ascending powers of θ\thetaθ, of the binomial expansion of L(θ)L(\theta)L(θ).
In a dual-sensor array, the combined response R(θ)R(\theta)R(θ) is given by R(θ)=(2−12θ)6+(2+12θ)6R(\theta) = \left(2 - \frac{1}{2}\theta\right)^6 + \left(2 + \frac{1}{2}\theta\right)^6R(θ)=(2−21θ)6+(2+21θ)6 Given that θ \theta\,θ is small enough that terms in θ4 \theta^4\,θ4 and higher powers of θ \theta\,θ may be neglected, show that R(θ)≈A+Bθ2R(\theta) \approx A + B\theta^2R(θ)≈A+Bθ2 where A A\,A and B B\,B are constants to be found.