Now equating real and imaginary part, we have cos ? = cos3 ? ? 3 cos ? sin" ? and sin ? = 3 cos" ? sin ? ? sin3 ? Assignment 1. Find the expansion for cos 4? and sin 4? in terms of cos ? 2. Simplify the following: i. (cos ? ? i sin ?) ? (cos 2? + i sin 2?) ii. (cos ? ? i sin ?)3 iii. (cos 2? ? i sin 2?)? ? (cos 3? ? i sin 3?)3 3. If z = cos ? + i sin ? express in terms of ? i. z + 1?z ii. z ? 1?z iii. ? + 1?z iv. ? ? 1?z SAMPLE QUESTIONS ON COMPLEX NUMBERS 1. Solve z? = ?2 + (2?3)i, giving the roots in the form r(cos ? + i sin ?). 2. If (1 + x)? = P? + P?x + P?x" + P?x3 + ..., prove that; i. P? ? P? + P? ? ... = 2^(??²) cos(n??4) ii. P? ? P? + P— ... = 2^(??²) sin(n??4) 3. The point P represents a complex number z on the argand diagram such that |z ? 6i| = 2|z ? 3|, show that as z varies, the locus of P is a circle, stating the radius and coordinate of the Centre of this circle. 4. Express sin? ? in terms of the form cos k? where is a positive integer. 5. Express sin3 ? in terms of the form sin k? where is a positive integer.
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To find the expansion for \( \cos 4 \theta \) and \( \sin 4 \theta \) in terms of \( \cos \theta \), we can use the double angle and triple angle formulas: \[ \begin{array}{l} \cos 4 \theta = 2 \cos^2 2 \theta - 1 = 2(2 \cos^2 \theta - 1)^2 - 1 \\ \sin 4 \theta = Show more…
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(a) Show that $w^{*}=\sqrt{z}$ has the values $$w_{1}=\sqrt{r}\left[\cos \frac{\theta}{2}+i \sin \frac{\theta}{2}\right].$$ $$\begin{aligned} w_{2} &=\sqrt{r}\left[\cos \left(\frac{\theta}{2}+\pi\right)+l \sin \left(\frac{\theta}{2}+\pi\right)\right] \\&=-w_{1}\end{aligned}.$$ (b) Obtain from (18) the often more practical formula $$\sqrt{z}=\pm[\sqrt{\frac{1}{2}(|z|+x)}+(\operatorname{sign} y) i \sqrt{\frac{x}{2}(|z|+x)}]$$ where sign $y=1$ If $y \geq 0, \operatorname{sign} y=-1$ if $y<0$ and all square roots of positive numbers are taken with positive sign. Hint: Use (10) in App. A3.1 with $x=\theta / 2$ (c) Find the square roots of $4 i, 16-30 i,$ and $9+8 \sqrt{7} /$ by both (18) and (19) and comment on the work involved. (d) Do some further examples of your own and apply a method of chocking your results.
Complex Numbers and Functions
Polar Form of Complex Numbers, Powers and Roots
Find the indicated roots. Express the results in rectangular form. (a) If $z=r(\cos \theta+i \sin \theta), z$ is not zero, and $n$ is a positive integer, use the result of Exercise 77 and DeMoivre's theorem to show that $z^{-n}=r^{-n}[\cos (-n \theta)+i \sin (-n \theta)]$ (b) If $z=r(\cos \theta+i \sin \theta)$ and $z$ is not zero, we define $z^{0}$ to be $1 .$ Show that $z^{0}=r^{0}(\cos 0+i \sin 0)$ (c) Finally, for any nonzero complex number $z=r(\cos \theta+i \sin \theta)$ and any integer $n,$ obtain the following generalization of DeMoivre's theorem: $$ z^{n}=r^{n}[\cos (n \theta)+i \sin (n \theta)] $$
Additional Topics in Trigonometry
Demoivre’s Theorem
Use the following relations to solve problem 1 and 2: z = x + jy Rectangular form z = r/ϕ Polar form (9.15) z = re^{jϕ} Exponential form The relationship between the rectangular form and the polar form is shown in Fig. 9.6, where the x axis represents the real part and the y axis represents the imaginary part of a complex number. Given x and y, we can get r and ϕ as r = √(x² + y²), ϕ = tan⁻¹(y/x) (9.16a) On the other hand, if we know r and ϕ, we can obtain x and y as x = r cos ϕ, y = r sin ϕ (9.16b) Thus, z may be written as z = x + jy = r/ϕ = r(cos ϕ + j sin ϕ) (9.17) Addition and subtraction of complex numbers are better performed in rectangular form; multiplication and division are better done in polar form. Given the complex numbers z₁ = x₁ + jy₁ = r₁/ϕ₁ z₂ = x₂ + jy₂ = r₂/ϕ₂ the following operations are important. Addition: z₁ + z₂ = (x₁ + x₂) + j(y₁ + y₂) (9.18a) Subtraction: z₁ - z₂ = (x₁ - x₂) + j(y₁ - y₂) (9.18b) Multiplication: z₁z₂ = r₁r₂/ϕ₁ + ϕ₂ (9.18c) Division: z₁/z₂ = r₁/r₂/ϕ₁ - ϕ₂ (9.18d) Reciprocal: 1/z = 1/r/-ϕ (9.18e) Square Root: √ z = √ r/ϕ/2 (9.18f) Complex Conjugate: z* = x - jy = r/-ϕ = re⁻ᴶᵠ (9.18g) Note that from Eq. (9.18e), 1/j = -j (9.18h) Alexander, Charles K. Fundamentals of electric circuits / Charles K. Alexander, Matthew N. O. Sadiku. — 5th ed.p. cm
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