so, you have your reference angle π/6
Now, sin2θ has period 2π/2 = π, so all solutions are the values
π/6 + nπ: π/6, 7π/6
5π/6 + nπ: 5π/6, 11π/6
where you just include the ones in [0,2π)
sin(2θ)=1/2
Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.) k = any integer
What I have so far: pi/12 + kpi, 5pi/12 + kpi
However, I do not know how to solve for the interval. Please help, thank you!
3 answers
θ = π / 12 + k π and θ = 5 π / 12 + k π
just put k = 0 , ±1 , ±2 , ±3... in this solutions
k = - 1
θ = π / 12 + k π = π / 12 - 1 ∙ π = π / 12 - π = π / 12 - 12 π / 12 = - 11 π / 12
θ = π / 12 + k π = 5 π / 12 - 1 ∙ π = 5 π / 12 - π = 5 π / 12 - 12 π / 12 = - 7 π / 12
θ = - 11 π / 12 and θ = - 7 π / 12 are less than 0, lie outside the interval [ 0 , 2 π ) and they are not solutions.
For k = - 2 , - 3 , - 4 ... θ is also less than 0, lie outside the interval and they are not solutions.
k = 0
θ = π / 12 + k π = π / 12 + 0 ∙ π = π / 12
θ = π / 12 + k π = 5 π / 12 + 0 ∙ π = 5 π / 12
k = 1
θ = π / 12 + k π = π / 12 + 1 ∙ π = π / 12 + π = π / 12 + 12 π / 12 = 13 π / 12
θ = π / 12 + k π = 5 π / 12 + 1 ∙ π = 5 π / 12 + π = 5 π / 12 + 12 π / 12 = 17 π / 12
k = 2
θ = π / 12 + k π = π / 12 + 2 π
θ = π / 12 + k π = 5 π / 12 + 2 π
θ = π / 12 + 2 π and θ = 5 π / 12 + 2 π are greater than 2 π, lie outside the interval [ 0 , 2 π ) and they are not solutions.
For k = 3 , 4 , 5 ... θ is also greater than 2 π, lie outside the interval and they are not solutions.
So the solutons are:
π / 12 , 5 π / 12 , 13 π / 12 , 17 π / 12
just put k = 0 , ±1 , ±2 , ±3... in this solutions
k = - 1
θ = π / 12 + k π = π / 12 - 1 ∙ π = π / 12 - π = π / 12 - 12 π / 12 = - 11 π / 12
θ = π / 12 + k π = 5 π / 12 - 1 ∙ π = 5 π / 12 - π = 5 π / 12 - 12 π / 12 = - 7 π / 12
θ = - 11 π / 12 and θ = - 7 π / 12 are less than 0, lie outside the interval [ 0 , 2 π ) and they are not solutions.
For k = - 2 , - 3 , - 4 ... θ is also less than 0, lie outside the interval and they are not solutions.
k = 0
θ = π / 12 + k π = π / 12 + 0 ∙ π = π / 12
θ = π / 12 + k π = 5 π / 12 + 0 ∙ π = 5 π / 12
k = 1
θ = π / 12 + k π = π / 12 + 1 ∙ π = π / 12 + π = π / 12 + 12 π / 12 = 13 π / 12
θ = π / 12 + k π = 5 π / 12 + 1 ∙ π = 5 π / 12 + π = 5 π / 12 + 12 π / 12 = 17 π / 12
k = 2
θ = π / 12 + k π = π / 12 + 2 π
θ = π / 12 + k π = 5 π / 12 + 2 π
θ = π / 12 + 2 π and θ = 5 π / 12 + 2 π are greater than 2 π, lie outside the interval [ 0 , 2 π ) and they are not solutions.
For k = 3 , 4 , 5 ... θ is also greater than 2 π, lie outside the interval and they are not solutions.
So the solutons are:
π / 12 , 5 π / 12 , 13 π / 12 , 17 π / 12
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