eliminate the parameter t from the following
x = 4 sin t y= 7 cos t
sketch the graph
t = sin-1 (x/4)
do the t for the other equation the same way, then equate the two results.
to graph it, I would stick with the original equations, and substitute some critical values for t such as 30,45,60,90º or pi/6,pi/4,pi/3,pi/2 etc if you are in radians.
To eliminate the parameter t from the equations x = 4 sin t and y = 7 cos t, we can use trigonometric identities.
1. Start by dividing both equations by their respective coefficients (4 and 7) to isolate the trigonometric functions.
x/4 = sin t
y/7 = cos t
2. Square both equations to get rid of the trigonometric functions:
(x/4)^2 = (sin t)^2
(y/7)^2 = (cos t)^2
3. Use the identity sin^2 t + cos^2 t = 1 to substitute back into the equations:
(x/4)^2 + (y/7)^2 = 1
4. Rearrange the equation in standard form:
(x/4)^2 + (y/7)^2 - 1 = 0
Now, let's sketch the graph.
The equation (x/4)^2 + (y/7)^2 - 1 = 0 represents an ellipse centered at the origin (0, 0) with semi-major axis of length 4 and semi-minor axis of length 7.
To sketch the graph:
1. Draw the x and y axes on your graph paper.
2. Plot the points on the x-axis with x = -4, 0, and 4.
3. Plot the points on the y-axis with y = -7 and 7.
4. Draw the ellipse passing through these points, making sure the distances from the center to each point on the ellipse are proportional to the semi-major and semi-minor axes.
The resulting graph will be an elliptical shape centered at the origin.