Why is {y|y > 1} the range of x = sin t, y = csc t, 0 < t < ∏ / 2 and the Cartesian equation y = 1 / x?
Also, why is {x|- 1 ≤ x ≤ 1} the domain of x = sin θ, y = cos 2θ and the Cartesian equation y = 1 - 2x^2?
Also, why is {x|- 1 ≤ x ≤ 1} the domain of x = sin θ, y = cos 2θ and the Cartesian equation y = 1 - 2x^2?
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These are parametric equations describing a certain path.
x = sin(t), y = 1/sin(t) = 1/x. But you need to recognise that for 0 < t < pi/2 then y lies anywhere between (1, infinity) so that y > 1. You can check by graphing this that x must lie between the interval (0, 1) but this can be determined by applying the same logic, that sin(t) has a range of (0, 1) over the given interval.
For the second graph, the range of sinθ is [-1, 1], so then x must lie within these values. You can use double angle formulas to find the Cartesian equation.
cos2θ = 1 - 2sin^2(θ) = 1 - 2x^2
x = sin(t), y = 1/sin(t) = 1/x. But you need to recognise that for 0 < t < pi/2 then y lies anywhere between (1, infinity) so that y > 1. You can check by graphing this that x must lie between the interval (0, 1) but this can be determined by applying the same logic, that sin(t) has a range of (0, 1) over the given interval.
For the second graph, the range of sinθ is [-1, 1], so then x must lie within these values. You can use double angle formulas to find the Cartesian equation.
cos2θ = 1 - 2sin^2(θ) = 1 - 2x^2