Question

sinx*sin2x*sin3x*sin4x.....upto sin12x=0. Find no. of solution lying in interval x€(0,π]​

Answer 1

Answer:

46 Solution

Step-by-step explanation:

sinx*sin2x*sin3x*sin4x.....upto sin12x=0

=> Sinx = 0

0r Sin2x = 0

   Sin3x = 0

   Sin4x = 0

   Sin5x = 0

   Sin6x = 0

  Sin7x = 0

   Sin8x = 0

  Sin9x = 0

  Sin10x = 0

  Sin11x = 0

  Sin12x = 0

Sinθ = 0  When θ = nπ    

x€(0,π]​

=> x = nπ   => x = π  

 2x = nπ   => x = π/2 , π

 3x = nπ => x = π/3 , 2π/3 , π

 4x = nπ => x = π/4 , π/2 , 3π/4 , π

 5x = nπ => x = π/5 , 2π/5 , 3π/5 , 4π/5 , π

 6x = nπ => x = π/6 , π/3 , π/2 , 2π/3 , 5π/6 , π

7x = nπ => x = π/7 , 2π/7 , 3π/7 , 4π/7 , 5π/7 , 6π/7 , π

8x = nπ => x = π/8 , π/4 , 3π/8 , π/2 , 5π/8 , 3π/4 , 7π/8 , π

9x = nπ => x = π/9 , 2π/9 , π/3 , 4π/9 , 5π/9 , 2π/3 , 7π/9 , 8π/9 , π

10x = nπ => x = π/10 , π/5 , 3π/10 , 2π/5 , π/2 , 3π/5 , 7π/10 , 4π/5 , 9π/10 , π

11x = nπ => x = π/11 , 2π/11 , 3π/11 , 4π/11 , 5π/11 , 6π/11 , 7π/11 , 8π/11 , 9π/11 , 10π/11 , π

12 x= nπ => x = π/12 , π/6 , π/4 , π/3 , 5π/12 , π/2 , 7π/12 , 2π/3 , 3π/4 , 5π/6 , 11π/12 , π

Number of unique solution other than π

5 , 7 , 11 are prime so they are unique

=4 + 6 + 10 = 20

12 has  11  (includes 2 , 3 , 4 , 6 solution)

10 has 4  more  ( excluding 2 & 5)

9 has  6 More  (excluding 3)

8 has  4  More ( excluding 2 & 4)

Total solution = 20 + 11 + 4 + 6 + 4 = 45

45 other than π

=> total = 46 Solution