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

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

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

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

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

6. Expansion of Sin and Cosine
Expand LHS using Binomial Theorem:

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

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

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

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

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

12. 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.

13. 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:

14. 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

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

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

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

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