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Trigonometric Form of Complex Numbers Unit 8: Section 6.5, Page 440-449 Warm up β’ Convert the following polar coordinates to rectangular β’ (4, 7π ) 6 β’ Convert the following rectangular coordinates to polar β’ (2,-3) β’ Hint: Radians!!! The Complex Plane β’ The complex plane is very similar to the x-y plane β’ Complex number z = x + iy β’ X-axis = real part β’ Y-axis = imaginary part β’ Complex numbers can also be written using trigonometry. Trig Form of Complex Nos. β’ z = x + iy β’ π = π πππ π½ + π πππ π½ β’ π= π₯ 2 + π¦ 2 tan π = β’ x = r cos ΞΈ π¦ π₯ y = r sin ΞΈ β’ Does this look familiar??? Itβs similar to converting between rectangular and polar coordinates! β’ Why?? The trig form makes it much easier to work with powers and roots of a complex number Converting: Complex to Trig β’ π= π₯ 2 + π¦ 2 tan π = π¦ π₯ x = r cos ΞΈ β’ Convert z = 6 - 6i to trig form β’ π = 62 + 62 = 2 β 36 = 6 2 β’ (6, -6) is in the 3rd Quadrant! 6 6 β’ tan πβ² = = 1 β’ πβ² = π 4 π 4 π=π+ = β’ π§ = 6 2 cos 7π 4 + π sin 7π 4 7π 4 or z = 6 2 πππ 7π 4 y = r sin ΞΈ Converting: Trig to Complex β’ π= π₯ 2 + π¦ 2 tan π = β’ Convert π§ = 3 cos 3π =3 2 3π 3 sin = 3 2 3π 2 π¦ π₯ + π sin β’ π₯ = 3 cos 0 =0 β’ π¦= β1 = β3 β’ z = (0, -3) x = r cos ΞΈ 3π 2 to complex. y = r sin ΞΈ Complex Numbers in Trig Form Multiplying & Dividing β’ π§1 π§2 = π1 π2 cos π1 +π2 + i sin π1 + π2 β’ π§1 π§2 = π1 π2 cos π1 β π2 + π sin π1 β π2 β’ Example: Find β’ π§1 π§2 = 3 4 3 4 πππ 4 cos β’ π§1 π§2 = 3 πππ 13π 12 π 3 π 3 + 4 πππ 3π 4 3π 4 + i sin π 3 + 3π 4 Complex Numbers in Trig Form Multiplying & Dividing (cont.) β’ π§1 π§2 = π1 π2 cos π1 +π2 + i sin π1 + π2 β’ π§1 π§2 = π1 π2 cos π1 β π2 + π sin π1 β π2 β’ Example: Find 24 8 24 πππ 300° 8 πππ 75° β’ π§1 π§2 = β’ π§1 π§2 = 3 cos 225 + π sin 225 cos 300 β 75 + π sin 300 β 75 =3 β 2 β 2 π 2 2 DeMoivreβs Theorem
