Honors Advanced Math Name__________________ Practice Test: Polar, Parametric, and Complex Numbers General instructions: Write a complete, fully explained solution to each problem, except where directions say otherwise. The quality of your responses will be a factor in grading. No calculators on this exam!

1. a. Sketch the graph of 3πθ = . No work need be shown.

b. Carefully sketch the graph of the following piecewise function. Show your work and include the coordinates of any important points.

⎩⎨⎧

<≤−

<≤−=

πθπθ

πθθ

2cos03)2cos(2

ifif

r

2. a. Prove that

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r1cisθ1 ⋅ r2cisθ2 = r1r2cis θ1 + θ2( ) .

b. Prove

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r1cisθ1r2cisθ2

=r1r2cis θ1 −θ2( )

3. Let z = 2 cis (7π/6) and w = i+− 3 .

a. Find wz 3 by doing calculations in polar mode. Leave your answer in polar form.

b. Find all the complex numbers that are 4th roots of w. Leave your answer in polar form.

4. Tommy is riding a Ferris wheel that turns counterclockwise at a rate of 1 revolution every 3 minutes. The radius of the Ferris wheel is 20 feet. At its lowest point, the Ferris wheel is 5 feet off the ground.

a. Suppose Tommy starts the ride in the 3 o'clock position. Write parametric equations describing the path that Tommy takes on the Ferris wheel.

b. Andrew stands 90 feet to the right of the base of the Ferris wheel and throws a ball at Tommy. Andrew releases the ball at a height of 6 feet at the exact time that Tommy starts his ride. Andrew's throw has an initial velocity of 78 feet/second and makes a

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76o angle with the horizontal. Write a pair of parametric equations describing the path of the ball. Hint: The height of the ball is given by the formula

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H = −16t2 + v0t + h0, where t is the time in seconds,

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v0 is the initial velocity in the upward direction, and

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h0 is the initial height.

c. Write an equation that you could graph to find the minimum distance between Tommy and the ball. Simply write the equation. You do not have to graph it.

5. This problem involves mixing

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a + bi and

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rcisθ notation. You may put your answers in either notation or you may use a combination of the two.

a. Find the five fifth roots of unity.

b. The answers you found in part a should be five points in the complex plane, evenly spaced, on the unit circle. Suppose these each of these points are translate right by 3 and up by 6i, so that they now lie on a circle of radius 1 centered at the number

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3+ 6i . Write the values of the five new numbers. (Note: Your answers may involve a combination of notations).

c. Consider the four numbers 0,

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cis(0o) ,

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2cis(45o) ,

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cis(90o) . Sketch these four numbers in the complex plane. What shape do they form?

d. Suppose that the four numbers from part c are replaced with the numbers 0,

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5cis(15o ) ,

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5 2cis(60o ) ,

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5cis(105o ) . What were the transformation(s) that were done to the four numbers from part c? Sketch the shape formed by the four new numbers.