Write a set of parametric equations in Quadrants I and IV for a semi-circle with a radius of 10.
I'm kind of new at parametrics, so this is what was given to me:
(x-h)^2 + (y-k)^2 = r^2
x^2 + y^2 = r^2
r = 10, y = rsint, x=rcost
So,
10cost + 10sint = 100
The answer is supposed to be
pi/2 < t < -pi/2
(actually greater than/less than or equal to)
My question is how do we get to the final solution, which is the pi/2 boundary. I know how to do the first part, I'm just not sure about the second...
Thank you!
To find the boundaries for the parameter t, we can consider the properties of a semi-circle.
In Quadrant I, the x-coordinate is positive, and the y-coordinate is positive. Using the parametric equations x = rcos(t) and y = rsin(t), we can substitute in the values for the radius (r = 10).
For Quadrant I:
x = 10cos(t)
y = 10sin(t)
Since the semi-circle is in Quadrant I, we know that the y-coordinate is positive (y > 0). Hence, 10sin(t) > 0.
Dividing both sides of 10sin(t) > 0 by 10, we have sin(t) > 0.
To find the boundary for t, we need to determine when sine is positive. In Quadrant I, sine is positive for values between 0 and π/2 (but not including π/2). Therefore, the boundary for t in Quadrant I is given by 0 < t < π/2.
In Quadrant IV, the x-coordinate is positive, and the y-coordinate is negative. Using the same parametric equations x = rcos(t) and y = rsin(t), we can again substitute in the radius (r = 10).
For Quadrant IV:
x = 10cos(t)
y = 10sin(t)
Since the semi-circle is in Quadrant IV, we know that the y-coordinate is negative (y < 0). Hence, 10sin(t) < 0.
Dividing both sides of 10sin(t) < 0 by 10, we have sin(t) < 0.
To find the boundary for t, we need to determine when sine is negative. In Quadrant IV, sine is negative for values between π/2 and π (but not including π). Therefore, the boundary for t in Quadrant IV is given by π/2 < t < π.
So, combining the boundaries for both Quadrant I and Quadrant IV, we have:
0 < t < π/2 (for Quadrant I)
π/2 < t < π (for Quadrant IV)
Hence, the final solution is pi/2 < t < -pi/2, where the ">"/"<" signs should be "<=" signs to include the endpoints.