x = sin(θ/2), y= cos(θ/2), −π ≤ θ ≤ π.
Eliminate the parameter to find a Cartesian equation of the curve.
clearly, x^2+y^2 = 1
The question is, how much of the circle is graphed?
θ x y
-π -1 0
0 0 1
π 1 0
we only get the top half of the circle.
To eliminate the parameter and find a Cartesian equation of the curve, we can use the trigonometric identity:
sin^2(θ/2) + cos^2(θ/2) = 1
Substituting x = sin(θ/2) and y = cos(θ/2) into the identity, we get:
x^2 + y^2 = 1
This is the equation of a circle in Cartesian coordinates with radius 1. So, the Cartesian equation of the curve is x^2 + y^2 = 1.
To eliminate the parameter in this case, we can use the trigonometric identity:
sin²(θ/2) + cos²(θ/2) = 1
Let's solve for sin²(θ/2) first:
sin²(θ/2) = 1 - cos²(θ/2)
Now substitute the given values of x and y:
x = sin(θ/2)
x² = sin²(θ/2)
Since sin²(θ/2) = 1 - cos²(θ/2), we can rewrite the equation as:
x² = 1 - cos²(θ/2)
Now, solve for cos²(θ/2):
cos²(θ/2) = 1 - x²
Substitute the value of y:
y = cos(θ/2)
y² = cos²(θ/2)
Since cos²(θ/2) = 1 - x², we can rewrite the equation as:
y² = 1 - x²
Therefore, by eliminating the parameter, the Cartesian equation of the curve is:
x² + y² = 1