Trig 2.3 Polar Coordinates Objective: to convert rectangular coordinates to polar and vice versa

Review Complex Plane Imagine that you are walking to the point 2 + 3i. Instead of walking 2 units right, turning 90⁰, and walking 3 more units, you want to take the nearest route. You need to have a direction and a distance to walk. What is the direction you need to take? How far should you walk?

Polar Coordinates Every polar coordinate has both an angle (direction) and a radius (distance). (r, θ) Partner Work – each group needs a balloon, two rubber bands, and a permanent marker. One person will be the coach, the other is the recorder.

Plot the Points (4, 0⁰) (2, 3π/2) (-2, 210⁰) (3, 135⁰) Now, on paper, plot the following points: (4, 30o) (-4, 225o) (2, -300o) (-3, -270o)

Polar Form of Complex Numbers The polar form or trigonometric form of the complex number a + bi is r(cos θ + i sin θ). To express -3 + 4i in polar form, first find the radius and the argument. r = √(32 + 42) θ = tan-1(4/-3) QII -3 + 4i = 5(cos 2.21 + i sin 2.21) = 5 cis 2.21 Express 2 – 2i√5 in polar form. Express 1 + √3i in polar form.

From Polar Form to Rectangular a = r cos θ and b = r sin θ Express 5 cis π/6 in rectangular form. x = 5 cos 30⁰ y = 5 sin 30⁰ x = 5 (√(3)/2) y = 5 (1/2) 5√3/2 + 5/2 i Express 10 cis 300⁰ in rectangular form. Express 4 cis 135o in rectangular form.

Practice Problems Assignment page 95 8, 9a, 10, 15 Find the magnitude and argument of each of the following numbers. 2√3 – 2i 4 cis 300o If z = 5 cis 75o and w = 2 cis 100o, find |zw| and arg(zw). Assignment page 95 8, 9a, 10, 15