Q. Let n be an add integer if `sin ntheta = sum_(r=0)^n(b_r)sin^rtheta`, for every value of theta then, a. `b_0 = 1, b_1 = 3` b. `b_0 = 0, b_1 = 1` c. `b_0 = -1, b_1 = 1` d. `b_0 = 0, b_1 = 2`

Options

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

B

Explanation

We have,
`sinn theta=underset(r=0)overset(n)(sum)b_(r)sin^(r)theta"for all"theta.`
Putting `theta=0,` we get ` 0=b_(0)`
Again, `sinn theta=underset(r=0)overset(n)(sum)b_(r)sin^(r)theta`
`implies(sinntheta)/(sintheta)=underset(r=0)overset(n)(sum)b_(r)(sintheta)r^(r-1)`
`impliesunderset(6to0)(lim)(sinntheta)/(sintheta)=underset(r=1)overset(n)(sum)b_(r)underset(thetato0)(lim)(sintheta)^(r-1)`
`impliesn =b_(1)`
Hence`,_(0)=0and b_(1)=n`