Question

$sinnx={\sum }_{r=0}^n{a}_{r}si{n}^{r}x$ where n is an odd natural number, then

• A. a₀ = 1, a₁ = 2n
• B. a₀ = 1, a₁ = n
• C. a₀ = 0, a₁ = n
• D. a₀ = 0, a₁ = -n

Answer 1

The correct option is C

a₀ = 0, a₁ = n

sin nx = Im$\left(e^{inx}\right)$ = Im$\left[{(cosx+isinx)}^n\right]$

= ${}^n{C}_{1}co{s}^{n-1}x.sinx-{}^n{C}_{3}co{s}^{n-3}x.si{n}^{3}x+{}^n{C}_{5}co{s}^{n-5}x.si{n}^{5}x+...............$

Since n is odd, let n = 2$\lambda$ + 1

$\Rightarrow$ ${}^n{C}_1(co{s}^{2}x{)}^{\lambda }sinx-{}^n{C}_3(co{s}^{2}x{)}^{\lambda -1}si{n}^{3}x+..........$

$\Rightarrow$ ${}^n{C}_1(1-si{n}^{2}x{)}^{\lambda }sinx-{}^n{C}_3(1-si{n}^{2}x{)}^{\lambda -1}si{n}^{3}x+..........$

$\Rightarrow$ ${}^n{C}_{1}sinx-({}^n{C}_1-{}^n{C}_1.{}^n{C}_3)si{n}^{3}x+.......$

$\Rightarrow$ $a_0=0,a_1=n$