If n1 denotes the maximum number of roots of sin θ=k1 in [0,2 π] and n2 denotes the maximum number of roots of cos θ=k2 in [0,2 π], then

Question

If n1 denotes the maximum number of roots of sin θ=k1 in [0,2 π] and n2 denotes the maximum number of roots of cos θ=k2 in [0,2 π], then (A) n1+n2=5 (B) n1+n2=4 (C) n1+n2=6 (D) n1+n2=3

Tardigrade Admin · Accepted Answer

sin θ=k1 has maximum roots if k1=0, so θ=π, π, 2 π cos θ=k2 has maximum roots if k2 π ± 1, so there are two values of θ. ∴ n1=3 and n2=2

Q. If $n_{1}$ denotes the maximum number of roots of $sin θ = k_{1}$ in $[0, 2 π]$ and $n_{2}$ denotes the maximum number of roots of $cos θ = k_{2}$ in $[0, 2 π]$ , then

$n_{1} + n_{2} = 5$

$n_{1} + n_{2} = 4$

$n_{1} + n_{2} = 6$

$n_{1} + n_{2} = 3$

$sin θ = k_{1}$ has maximum roots if $k_{1} = 0$ ,
so $θ = π, π, 2 π cos θ = k_{2}$ has maximum roots if $k_{2} π \pm 1$ ,
so there are two values of $θ$ .
$∴ n_{1} = 3$ and $n_{2} = 2$

Q. If n1​ denotes the maximum number of roots of sinθ=k1​ in [0,2π] and n2​ denotes the maximum number of roots of cosθ=k2​ in [0,2π], then

n1​+n2​=5

n1​+n2​=4

n1​+n2​=6

n1​+n2​=3