Understanding Parametric Equations for the Unit Circle

The video tutorial explores parametric equations for the unit circle, emphasizing the need for these equations to satisfy the rectangular equation x^2 + y^2 = 1 and cover all x and y values. It discusses the trigonometric identities involved and verifies various parametric equations through substitution and graphical methods. The tutorial also highlights common errors and corrects misconceptions about parametric equations representing the unit circle.

What is the rectangular equation for the unit circle?

• x^2 + y^2 = 1

• x^2 - y^2 = 1

• x^2 + y^2 = 0

• x^2 - y^2 = 0

Which of the following must parametric equations satisfy to represent the unit circle?

• They must be linear equations.

• They must have a constant interval for T.

• They must satisfy the rectangular equation.

• They must only cover positive x and y values.

Why do the parametric equations x = 2cos(T) and y = 2sin(T) not represent the unit circle?

• They have a radius of 2, not 1.

• They are not trigonometric functions.

• They do not satisfy the rectangular equation.

• They only cover half the circle.

What happens when you graph the parametric equations x = cos(3T) and y = sin(3T)?

• They only cover the top half of the circle.

• They do not form a closed shape.

• They trace the unit circle three times.

• They form an ellipse.

What is the effect of the coefficient in the parametric equations x = cos(3T) and y = sin(3T)?

• It makes the circle smaller.

• It changes the radius to 3.

• It changes the period of the circle.

• It makes the circle larger.

Why do the parametric equations x = T and y = sqrt(1 - T^2) not represent the full unit circle?

• They only cover the top half of the circle.

• They do not satisfy the rectangular equation.

• They only cover the bottom half of the circle.

• They are not trigonometric functions.

Which parametric equations correctly represent the unit circle?

• x = T, y = sqrt(1 - T^2)

• x = cos(2T), y = sin(2T)

• x = 2cos(T), y = 2sin(T)

• x = cos(T), y = sin(T)

What is the result of graphing x = sin(T) and y = cos(T)?

• A line is formed.

• An ellipse is formed.

• A unit circle is formed.

• A parabola is formed.

Which identity is used to verify parametric equations for the unit circle?

• cos(T) - sin(T) = 1

• cos^2(T) + sin^2(T) = 1

• cos(T) + sin(T) = 1

• cos^2(T) - sin^2(T) = 1

What is the effect of interchanging sine and cosine in parametric equations?

• The circle is not formed.

• The circle becomes a line.

• The unit circle is still traced.

• The circle becomes an ellipse.