Here are the solutions to questions 1a and 1b. 1a. Find the modulus and argument of For a complex number $Z = x + yi$: The modulus is $|Z| = \sqrt{x^2 + y^2}$. The argument is $\arg(Z) = \theta$, where $\tan\theta = \frac{y}{x}$, and $\theta$ is chosen based on the quadrant of $(x, y)$. i) $Z = 4 + 3i$ Step 1: Identify the real and imaginary parts. Here, $x = 4$ and $y = 3$. Step 2: Calculate the modulus. $$ |Z| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $$ Step 3: Calculate the argument. $$ \tan\theta = \frac{3}{4} $$ Since $x > 0$ and $y > 0$, $\theta$ is in the first quadrant. $$ \theta = \arctan\left(\frac{3}{4}\right) \approx 36.87^\circ \text{ or } 0.6435 \text{ radians} $$ The modulus is $\boxed{5}$ and the argument is $\boxed{\arctan\left(\frac{3}{4}\right)}$. ii) $Z = 1 + i$ Step 1: Identify the real and imaginary parts. Here, $x = 1$ and $y = 1$. Step 2: Calculate the modulus. $$ |Z| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} $$ Step 3: Calculate the argument. $$ \tan\theta = \frac{1}{1} = 1 $$ Since $x > 0$ and $y > 0$, $\theta$ is in the first quadrant. $$ \theta = \arctan(1) = \frac{\pi}{4} \text{ or } 45^\circ $$ The modulus is $\boxed{\sqrt{2}}$ and the argument is $\boxed{\frac{\pi}{4}}$. iii) $Z = \sin\theta + i\cos\theta$ Step 1: Identify the real and imaginary parts. Here, $x = \sin\theta$ and $y = \cos\theta$. Step 2: Calculate the modulus. $$ |Z| = \sqrt{(\sin\theta)^2 + (\cos\theta)^2} = \sqrt{\sin^2\theta + \cos^2\theta} $$ Using the identity $\sin^2\theta + \cos^2\theta = 1$: $$ |Z| = \sqrt{1} = 1 $$ Step 3: Calculate the argument. The standard polar form of a complex number is $r(\cos\phi + i\sin\phi)$. We have $Z = \sin\theta + i\cos\theta$. Using the trigonometric identities $\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right)$ and $\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$: $$ Z = \cos\left(\frac{\pi}{2} - \theta\right) + i\sin\left(\frac{\pi}{2} - \theta\right) $$ Comparing this to the polar form, the argument $\phi$ is $\frac{\pi}{2} - \theta$. The modulus is $\boxed{1}$ and the argument is $\boxed{\frac{\pi}{2} - \theta}$. 1b. Evaluate The powers of $i$ follow a cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. This cycle repeats every 4 powers. To evaluate $i^n$, find the remainder of $n$ when divided by 4. i) $i^{20}$ Step 1: Divide the exponent by 4. $$ 20 \div 4 = 5 \text{ with a remainder of } 0 $$ Step 2: Use the remainder to find the value. Since the remainder is 0, $i^{20}$ is equivalent to $i^4$. $$ i^{20} = i^4 = 1 $$ The value is $\boxed{1}$. ii) $Z = -1 + i^2$ Step 1: Substitute the value of $i^2$. We know $i^2 = -1$. $$ Z = -1 + (-1) $$ Step 2: Simplify the expression. $$ Z = -1 - 1 = -2 $$ The value is $\boxed{-2}$. iii) $i^{-4}$ Step 1: Rewrite the expression with a positive exponent. $$ i^{-4} = \frac{1}{i^4} $$ Step 2: Substitute the value of $i^4$. We know $i^4 = 1$. $$ \frac{1}{i^4} = \frac{1}{1} = 1 $$ The value is $\boxed{1}$. iv) $i^5$ Step 1: Divide the exponent by 4. $$ 5 \div 4 = 1 \text{ with a remainder of } 1 $$ Step 2: Use the remainder to find the value. Since the remainder is 1, $i^5$ is equivalent to $i^1$. $$ i^5 = i^1 = i $$ The value is $\boxed{i}$.