CHAPTER 1 COMPLEX NUMBERS EXPANSION FOR COS AND SINE LOCI IN COMPLEX NUMBER

Expansion of Sin and Cosine Example 1.19: State in terms of cosines. De Moivre’s Theorem? Finding Roots?? Should I just expand it??

Expansion of Sin and Cosine Theorem 5: If , then: Theorem 6: (Binomial Theorem) If , then:

Expansion of Sin and Cosine Example 1.20 Expand using binomial theorem, then write in standard form of complex number: Answer:

Expansion of Sin and Cosine Example 1.19: State in terms of cosines. Solution: By applying Theorem 5 and Binomial Theorem Theorem 5

Expansion of Sin and Cosine Expand LHS using Binomial Theorem:

Expansion of Sin and Cosine Expand RHS: Equate (2) and (3):

Expansion of Sin and Cosine Example 1.21: By using De Moivre’s theorem and Binomial theorem, prove that:

Expansion of Sin and Cosine Solution: By applying DMT: Expand LHS using Binomial theorem:

Expansion of Sin and Cosine Equate (1) and (2), then compare: Real part: Imaginary part:

Expansion of Sin and Cosine Example 1.22: Using appropriate theorems, state the following in terms of sine and cosine of multiple angles : Answer:

Loci in the Complex Number Since any complex number, z = x+iy correspond to point (x,y) in complex plane, there are many kinds of regions and geometric figures in this plane can be represented by complex equations or inequations. Definition 1.9 A locus in a complex plane is the set of points that have specified property. A locus in a complex plane could be a straight line, circle, ellipse and etc.

Loci in the Complex Number Example 1.22: Equation of circle with center at the origin and radius, r y x P on circumference: P outside circle: P inside circle:

Loci in the Complex Number Example 1.23: What is the equation of circle in complex plane with radius 2 and center at 1+i Solution: y x Distance from center to any point P must be the same

Loci in the Complex Number Example 1.23: Find the equation of locus if:

Loci in the Complex Number Solution: * Locus eq : A straight line eq with m = -2

Loci in the Complex Number y x Distance from point (0,-1) and (2,0) to any point P must be the same

Loci in the Complex Number Example 1.24: Find the equation of locus if: