microsoft/qdk
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katas/content/complex_arithmetic/cartesian_to_polar/solution.md
24lines · modecode
| 1 | We need to calculate the $r$ and $\theta$ values as seen in the complex plane. |
| 2 | $r$ should be familiar to you already, since it is the modulus of a number (exercise 6): |
| 3 | |
| 4 | $$ r = \sqrt{a^2 + b^2} $$ |
| 5 | |
| 6 | $\theta$ can be calculated using trigonometry: since we know that the polar and the Cartesian forms of the number represent the same value, we can write |
| 7 | |
| 8 | $$ re^{i \theta} = a + bi $$ |
| 9 | |
| 10 | Euler's formula allows us to express the left part of the equation as |
| 11 | |
| 12 | $$ re^{i \theta} = r \cos \theta + i r \sin \theta $$ |
| 13 | |
| 14 | For two complex numbers to be equal, their real and imaginary parts have to be equal. This gives us the following system of equations: |
| 15 | |
| 16 | $$ \begin{cases} a = r \cos \theta \\ b = r \sin \theta \end{cases} $$ |
| 17 | |
| 18 | To calculate $\theta$, we can divide the second equation by the first one to get |
| 19 | |
| 20 | $$ \tan \theta = \frac{b}{a} $$ |
| 21 | |
| 22 | $$ \theta = \arctan \left(\frac{b}{a}\right) $$ |
| 23 | |
| 24 | @[solution]({"id": "complex_arithmetic__cartesian_to_polar_solution", "codePath": "Solution.qs"}) |
| 25 | |