A specialized camera sensor captures light in a region shaped like a circular sector OAB OAB\,OAB with radius r r\,r and central angle θ \theta\,θ radians. The point M M\,M is the midpoint of the radius OAOAOA. A point P P\,P lies on the radius OB OB\,OB such that MP MP\,MP is perpendicular to OBOBOB.
Given that the area of the sector OAB OAB\,OAB is exactly 6 times the area of the triangle OMPOMPOMP, show that 4θ=3sin2θ4\theta = 3\sin 2\theta4θ=3sin2θ.
Use a sign change method to show that a solution to the equation 4θ=3sin2θ4\theta = 3\sin 2\theta4θ=3sin2θ lies in the interval 0.7<θ<0.80.7 < \theta < 0.80.7<θ<0.8.
The Newton-Raphson method is used to find an approximate solution to 4θ−3sin2θ=04\theta - 3\sin 2\theta = 04θ−3sin2θ=0. Using θ1=0.8\theta_1 = 0.8θ1=0.8 as a first approximation, find the value of θ2 \theta_2\,θ2 to three decimal places.
Explain why choosing a first approximation where cos2θ=23\displaystyle \cos 2\theta = \frac{2}{3}cos2θ=32 would fail to find a solution using this method.