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Question 1 Show that \( \cos 4 \theta = 8 \cos^4 \theta – 8 \cos^2 \theta + 1 \). (a) By considering the real part of \( z^4 \), prove \( \cos 4\theta = \cos^4 \theta – 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta \) by letting \( z = \cos \theta + i […]
Complex Numbers with Vector Addition is obtained using the sides of triangles, the sum of sides of two sides of a triangle is greater than the other side. Worked Examples of Complex Numbers with Vector Addition Suppose \( \displaystyle 0 \lt \alpha, \ \beta \lt \frac{\pi}{2} \) and define complex numbers \(z_n\) by $$z_n = […]
Locus of Complex Numbers is obtained by letting ( z = x+yi ) and simplifying the expressions. Operations of modulus, conjugate pairs and arguments are to be used for determining the locus of complex numbers. Question Let \( z = x + iy \) be a complex number such that \(|z-1|^2 – |z-3|^2 = 1 […]
Regions defined by complex numbers \( \displaystyle z = x + yi \) where \(x\) and \(y\) are real numbers, can be drawn using inequalities for complex number regions. Worked Example of Complex Number Regions Find the inequality of complex region \( \displaystyle \Bigg|1+\frac{1}{z}\Bigg| \le 1 \). \( \begin{aligned} \displaystyle \require{color} \frac{|z+1|}{|z|} &\le 1 \\ […]