(a)

The parametric equations are and .

Find the rectangular equation.

Consider .

Squaring on each side.

.

Consider .

Squaring on each side.

.

Trigonometric identity: .

Substitute and .

The rectangular equation is .

(i)

Graph:

Graph the parametric equations and .

.

Observe the graph:

The domain of the graph is .

It is circle with Anticlockwise orientation and curve is smooth and continuous on the interval.

The range is .

(ii)

Graph:

Graph the parametric equations and .

.

Observe the graph:

The domain of the graph is .

It is circle with clockwise orientation and curve is smooth and continuous on the interval.

The range is .

(b)

Observe the set of graphs of parametric equations:

The parametric equations from (a)-(b) are similar and their rectangular equation is .

The curves are identical but orientation is opposite to each other.

(c)

Observe the set of parametric equations:

As the sign of changes, the orientation also changes.

Conjecture: As the sign of is reversed, the orientation will be reversed.

(d)

Let another set of parametric equation be and .

Graph:

Graph the parametric equations and .

.

Graph:

Graph the parametric equations and .

.

Observe the parametric equations and graphs:

As the sign of is reversed, the orientation will be reversed.

(a)

(i)

The domain of the graph is .

It is circle with Anticlockwise orientation and curve is smooth and continuous on the interval.

The range is .

(ii)

The domain of the graph is .

It is circle with clockwise orientation and curve is smooth and continuous on the interval.

The range is .

(b) The curves are identical but orientation is opposite to each other.

(c) As the sign of is reversed, the orientation will be reversed.

(d) Set of parametric equations are , and , .