Question

Let n be an odd integer. If $\sin n\theta =\sum _{r=0}^H{b}_r{\sin }^r\theta$ for all real $\theta$ then

• A. $b_0=1,b_1=3$
• B. $b_0=0,b_1=n$
• C. $b_0=-1,b_1=n$
• D. $b_0=0,b_1=n^2-3n-3$

Answer 1

Answer: (B)

$b_0=0,b_1=n$

$\sin \left(n\theta \right)=\sum _{r=1}^n{b}_R{\sin }^r\theta$

For $n=0$

$\sin 0=b^0{\sin }^0\theta$

$⟹b_0=0$

For $n=1$

$\sin \left(\theta \right)=b_0{\sin }^0\theta +b_1{\sin }^1\theta$

$\Rightarrow \sin \theta =b_0+b_1\sin \theta$

$\Rightarrow \sin \theta =0+b_1\sin \theta$

$\Rightarrow b_1=1$