find all solutions to the equation √3 csc(2theta)=-2
Would the answer be
π/6 + 2πn, 5π/6 +2πn or π/6 + πn, 5π/6 +πn?
or neither?
sqrt ( 3 ) * csc ( 2 theta ) = - 2 Divide both sides by sqrt ( 3 )
csc ( 2 theta ) = - 2 / sqrt ( 3 )
Take the inverse cosecant of both sides.
2 theta = - pi / 3
and
2 theta = 4 pi / 3
[ Becouse csc ( pi / 3 ) = 2 / sqrt ( 3 ) , and csc ( 4 pi / 3 = 2 / sqrt ( 3 ) ]
Divide both sides by 2
theta = - pi / 6
and
theta = 2 pi / 3
Period of csc ( theta ) = 2 pi
So period of 2 theta = 2 pi / 2 = pi
Final solutions :
theta = n pi + 2 pi / 3 = pi ( n + 2 / 3 )
and
theta = n pi - pi / 6 = pi ( n - 1 / 6 )
To find all the solutions to the equation √3 csc(2θ) = -2, where θ is an angle, we need to simplify the equation and then solve for θ.
Step 1: Simplify the equation
First, let's find the reciprocal of csc(2θ), which is just sin(2θ).
√3 * sin(2θ) = -2
Step 2: Solve for sin(2θ)
Divide both sides by √3:
sin(2θ) = -2/√3
Step 3: Find the reference angle
To find the reference angle, we need to evaluate the inverse sin of -2/√3. This gives us the angle in the first quadrant that has the same sine value as -2/√3.
sin^(-1)(-2/√3) ≈ -0.9828
Since sin is negative in the third and fourth quadrants, let's subtract -0.9828 from π:
π - 0.9828 ≈ 2.1584
So the reference angle is approximately 2.1584.
Step 4: Find the solutions in the interval [0, 2π)
To find the solutions in the given interval, we can add multiples of the period of sin(2θ), which is π.
θ = (2.1584 + πn)/2, where n is an integer
Simplifying this expression, we get:
θ = 1.0792 + (π/2)n
Therefore, the solutions to the equation √3 csc(2θ) = -2 in the interval [0, 2π) are:
θ = 1.0792 + (π/2)n, where n is an integer.
So the answer would be neither of the options you provided.