This figure shows concentric circles of radii 3 and 5, respectively, centered at the origin of an xy-coordinate system. A ray from the center at an angle of t degrees to the x-axis cuts the two circles at points A and B. From A, a vertical line is drawn; and from B, a horizontal line is drawn. These two lines intersect at point P on an ellipse.
For the value of t shown, draw a right triangle (on your paper) for which one of the legs equals the x-coordinate of point P. Use the result to write an equation for x as a function of t.
For the value of t shown, draw a right triangle (on your paper) for which one of the legs equals the y-coordinate of point P. Use the result to write an equation for y as a function of t.
Diagram can be found blondebeliever.tumblr.[com]/precalc (on my blog) under question 2!
x = 5cos(t)
y = 3sin(t)
To draw a right triangle for one of the legs equal to the x-coordinate of point P, follow these steps:
1. Draw a line segment from the origin (center of the circles) at an angle of t degrees to the x-axis. This line represents the ray from the center.
2. Extend this line until it intersects the outer circle of radius 5 at point A.
3. Draw a vertical line from point A until it intersects the horizontal line drawn from point B.
4. Label the point of intersection between the vertical line from A and the horizontal line from B as point P.
5. Connect point P to the origin with a straight line segment.
6. With point P as the apex, draw a perpendicular line to the x-axis, intersecting it at point Q.
7. The leg of the right triangle that represents the x-coordinate of point P is the segment PQ.
8. Mark the length of PQ on your paper.
To write an equation for x as a function of t, observe the right triangle you drew:
- The segment PQ represents the x-coordinate of point P on the ellipse.
- The hypotenuse, represented by the line segment connecting the origin to point P, is the radius 3 of the inner circle.
- The angle between the hypotenuse and the x-axis is t degrees.
Using trigonometry, we can determine the relationship between the x-coordinate (PQ) and the given angle t:
cos(t) = PQ/3
Rearranging the equation gives:
PQ = 3cos(t)
Thus, the equation for x as a function of t is x = 3cos(t).
To draw a right triangle for one of the legs equal to the y-coordinate of point P, follow these steps:
1. Follow steps 1-4 from the previous section to draw the ray, points A, B, and point P on the ellipse.
2. Connect point P to the origin with a straight line segment.
3. With point P as the apex, draw a perpendicular line to the y-axis, intersecting it at point R.
4. The leg of the right triangle that represents the y-coordinate of point P is the segment PR.
5. Mark the length of PR on your paper.
To write an equation for y as a function of t, observe the right triangle you drew:
- The segment PR represents the y-coordinate of point P on the ellipse.
- The hypotenuse, represented by the line segment connecting the origin to point P, is the radius 3 of the inner circle.
- The angle between the hypotenuse and the x-axis is t degrees.
Using trigonometry, we can determine the relationship between the y-coordinate (PR) and the given angle t:
sin(t) = PR/3
Rearranging the equation gives:
PR = 3sin(t)
Thus, the equation for y as a function of t is y = 3sin(t).
Remember to substitute the correct values of t into the equations for x and y in order to find the coordinates of point P on the ellipse.