consider an angle in standard position with r=12 centimeters. write a short paragraph describing the change in magnitudes of x,y,sine theta,cosine theta, and tan theta as theta increases continually from zero degrees to ninety degrees.

To analyze the change in magnitudes of x, y, sine theta, cosine theta, and tan theta as theta increases continually from zero degrees to ninety degrees for an angle in standard position with r = 12 centimeters, we need to understand the relationships between these values.

When the angle theta is equal to zero degrees, the point lies on the positive x-axis. Therefore, the magnitude of x is equal to 12 centimeters, while the magnitude of y and sine theta are both zero. The magnitude of cosine theta is 1, and the magnitude of tan theta is also zero since tan theta is equal to sine theta divided by cosine theta.

As theta increases from zero degrees, the point starts moving counterclockwise along the circumference of a circle with radius r = 12 centimeters. The magnitude of x will start decreasing gradually while the magnitude of y will start increasing. Sine theta represents the magnitude of y, so as theta increases, the magnitude of sine theta will increase as well.

The magnitude of cosine theta, which represents the magnitude of x, will decrease as theta increases. This happens because as the angle increases, the point moves away from the positive x-axis and towards the negative y-axis. Since cosine theta represents the x-coordinate, its magnitude will decrease.

The magnitude of tan theta, which is equal to sine theta divided by cosine theta, will also change. However, for values of theta less than 90 degrees, tan theta will remain zero. Once theta reaches 90 degrees, where the angle is a right angle, the magnitude of tan theta becomes undefined since cosine theta equals zero.

In summary, as theta increases from zero to 90 degrees in an angle in standard position with r = 12 centimeters, the magnitudes of x and cosine theta will decrease gradually, while the magnitudes of y, sine theta, and tan theta (when defined) will increase.