When learning about De Moivre's Theorem for complex roots in A-Level Maths, people often make some common mistakes. I've experienced this myself, so I want to share these mistakes to help you avoid confusion and get the right answers more quickly.

1. Forgetting to Change to Polar Form

One big mistake is not changing complex numbers into polar form first. De Moivre's Theorem works best when complex numbers are written as $r(\cos \theta + i\sin \theta)$. Here, $r$ is the length, and $\theta$ is the angle. If you jump right into calculations using the standard form (like $a + bi$), you might get lost. Always change the complex number to polar form before using the theorem.

2. Errors with the Length and Angle

When converting to polar form, many students mess up finding the length $r$ or the angle $\theta$. The length is just $r = \sqrt{a^2 + b^2}$, and the angle might involve using $\tan^{-1}(b/a)$. Don’t forget that you might need to adjust the angle based on what quadrant the complex number is in. A quick drawing of the complex plane can help you understand this better and avoid mistakes with the signs of $r$ and the value of $\theta$.

3. Confusing Powers and Roots

It's very important to remember the difference between finding powers and roots with De Moivre's Theorem. For powers, you use the theorem like this: $z^n = r^n(\cos(n\theta) + i\sin(n\theta))$. But when you’re finding roots, it’s not just dividing $r$ and $\theta$ by $n$. Each root can have several angles, written as $\frac{2k\pi}{n}$, where $k = 0, 1, ..., n-1$. Make sure to find all possible roots when needed!

4. Missing Some Roots

When calculating complex roots, especially for higher-order roots, don’t forget to find all $n$ different solutions. A common mistake is only finding one root and stopping. For example, if you need to find the cube roots of a complex number, make sure to find all three different angles. Increment $k$ in $\theta_k = \frac{\theta + 2k\pi}{n}$ for $k = 0, 1, 2$. It might take some extra effort, but it’s really important!

5. Angle Measurement Errors

Another mistake is forgetting to keep angles in the right forms. Depending on what you are working on, you might need to use degrees or radians. Make sure whenever you add, subtract, or calculate angles, they all use the same measurement to avoid mixing radians and degrees. This might seem small, but just one mistake can mess up the whole problem.

6. Forgetting to Simplify

Finally, be careful not to skip simplifying your final answer. After you get an answer in polar form, remember to convert it back to standard form if asked. This can include simplifying trigonometric expressions. Being clear and neat in your answers not only impresses teachers but also helps solidify your understanding.

Conclusion

Learning from others' mistakes is a great way to master De Moivre’s Theorem. By keeping an eye out for these common errors—like not changing to polar form, making mistakes with length and angles, confusing powers and roots, missing some roots, mixing angle measurements, and forgetting to simplify—you can make your work easier and feel more confident. Practice these ideas through past papers or review exercises, and you’ll tackle complex numbers with clarity and confidence. Happy studying!