Question

sin theta +cosec theta =2 then find 1/sin1000 theta +sin1000 theta

Answer 1

sin Θ + cos Θ = 2
=> ( sin Θ + cos Θ )² = 2² = 4
=> 2 sinΘ cosΘ = 3
=> (sinΘ - cosΘ)² = 1-3 = -2
=> sinΘ - cosΘ = √2i
=>2sinΘ = 2+√2i ; 2cosΘ = 2-√2i
=>Θ = 0.544 + 1.276i
=> 1000Θ = 544 + 1276i

=>sin 1000Θ = sin544 cos1276i + sin1276i cos544
=> sin1000Θ = (-0.241) * 7.223e553i + 7.223e553i * (-0.970)
=> sin 1000Θ = -1.74e553i - 7.006e553i = -8.746e553 i

Hence, 1/sinΘ + sinΘ = (-8.746e553 i)^-1 + -8.746e553 i