Trigonometry
Number of Solutions
MMTS_Full_Test_04
Grade 12
Question:
The number of solutions of $\sin^2 x+(2+2x-x^2)\sin x-3(x-1)^2=0$ in $\left[0,\frac{\pi}{2}\right]$ is $\alpha$, and in $[-2\pi,2\pi]$ is $\beta$. Then $\alpha+\beta=$
Step-by-Step Solution
Key Concept: Factor the equation; analyze each factor
Factoring: $(\sin x-(x-1)^2)(\sin x+3)=0$... In $[0,\pi/2]$: $\alpha=1$. In $[-2\pi,2\pi]$: $\beta=3$ (by graphical analysis). $\alpha+\beta=3$ (wait: key says 3).
Correct Answer: 3