Question

The most general values of $\theta$ for which $\sin \theta -\cos \theta =\underset{a\in R}{min}(1,a^2-6a+11)$ are given by

• A. $n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4},n\in Z$
• B. $n\pi +(-1{)}^n\frac{\pi }{4}+\frac{\pi }{4},n\in Z$
• C. $2n\pi +\frac{\pi }{2},n\in Z$
• D. $n\pi +\frac{\pi }{2},n\in Z$

Answer 1

Answer: (C)

$2n\pi +\frac{\pi }{2},n\in Z$

$min(1,a^2-6a+11)=min(1,(a-3{)}^2+2\left)\right)=1$

$\Rightarrow \sin \theta -\cos \theta =1$

$\Rightarrow \frac{2\tan \frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}-\frac{1-{\tan }^2\frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}=1$

$\Rightarrow \frac{x^2+2x-1}{1+x^2}=1$

$\Rightarrow x=1$

$\Rightarrow \tan \frac{\theta }{2}=1$

So $\frac{\theta }{2}=n\pi +\frac{\pi }{4}$