Question
determine the values of theta where sin theta =-3square root over 2 and 0 greater/equal sign thethat greater/equal 2pi
Answers
GPT 3.5
The square root of 2 is a positive number, so it is not possible for the sine of an angle to equal -3 times the square root of 2. Therefore, there are no values of theta that satisfy the given condition.
oobleck
the question should be read as
where does sinθ = -√3/2 where 0 <= θ < 2π
we know that sin π/3 = √3/2, and
sinθ < 0 in QIII and QIV
so sinθ = -√3/2 if θ = 4π/3 or 5π/3
where does sinθ = -√3/2 where 0 <= θ < 2π
we know that sin π/3 = √3/2, and
sinθ < 0 in QIII and QIV
so sinθ = -√3/2 if θ = 4π/3 or 5π/3
GPT 3.5
Apologies for the misunderstanding.
Indeed, we can consider the unit circle to find the values of theta where sin theta = -√3/2, where 0<=theta<2π.
Recall that sin(pi/3) = √3/2.
On the unit circle, sin theta corresponds to the y-coordinate of a point on the circle. Since sin theta is negative (sin theta < 0) in Quadrant III and Quadrant IV, we need to find the angles in these quadrants that have the same absolute value as sin(pi/3).
In Quadrant III, the angle theta that satisfies sin theta = -√3/2 is θ = 2π - π/3 = 5π/3.
In Quadrant IV, the angle theta that satisfies sin theta = -√3/2 is θ = 2π + π/3 = 7π/3.
Therefore, the values of theta where sin theta = -√3/2, and 0<=theta<2π, are θ = 5π/3 and θ = 7π/3.
Indeed, we can consider the unit circle to find the values of theta where sin theta = -√3/2, where 0<=theta<2π.
Recall that sin(pi/3) = √3/2.
On the unit circle, sin theta corresponds to the y-coordinate of a point on the circle. Since sin theta is negative (sin theta < 0) in Quadrant III and Quadrant IV, we need to find the angles in these quadrants that have the same absolute value as sin(pi/3).
In Quadrant III, the angle theta that satisfies sin theta = -√3/2 is θ = 2π - π/3 = 5π/3.
In Quadrant IV, the angle theta that satisfies sin theta = -√3/2 is θ = 2π + π/3 = 7π/3.
Therefore, the values of theta where sin theta = -√3/2, and 0<=theta<2π, are θ = 5π/3 and θ = 7π/3.