Use details/summary tag for examples

mechanical/mmme1026_maths_for_engineering.md

@@ -19,8 +19,8 @@ tags: [ uni, nottingham, mmme1026, maths, complex_numbers ]
 
   Where:
 
-  - $x$ is the real part of $z$ (Re($z$))
+  - $x$ is the real part of $z$ (which you may seen written as $\Re(z) = x$ or Re$(z) = x$)
-  - $y$ is the imaginary part of $z$(Im($z$))
+  - $y$ is the imaginary part of $z$ (which you may seen written as $\Im(z) = y$ or Im$(z) = y$)
 
 - Two complex numbers are equal if both their real and imaginary parts are equal
 
@@ -146,15 +146,23 @@ $$e^{i\theta} = \cos\theta + i\sin\theta$$
 
   **Note**: $$\bar z = r\cos\theta - ir\sin\theta = re^{-i\theta}$$
 
+<details>
+<summary>
 ### Example 1
 
 Write $z = -1 + i$ in exponential form
 
+</summary>
+
 > $\arg z = \frac {3\pi} 4$
 > $|z| = \sqrt 2$
 > 
 > So $z = \sqrt2e^{i\frac{3\pi} 4}$
 
+</details>
+
+<details>
+<summary>
 ### Example 2
 
 The equations for a mechanical vibration problem are found to have the following mathematical
@@ -162,6 +170,8 @@ solution:
 
 $$z(t) = \frac{e^{i\omega t}}{\omega_0^2-\omega^2 + i\gamma}$$
 
+</summary>
+
 where $t$ represents time and $\omega$, $\omega_0$ and $\gamma$ are all positive real physical
 constants.
 Although $z(t)$
@@ -200,6 +210,8 @@ b. Hence find the constants $b$ and $\varphi$ such that
      y(t) &= \frac 1 a \sin(\omega t - \delta) \\
    > \end{align*}
 
+</details>
+
 ## Products of Complex Numbers
 
 Suppose we have 2 complex numbers:
@@ -222,22 +234,35 @@ z_3 = z_1 z_2 &= (r_1e^{i\theta_1})(r_2e^{i\theta_2}) \\
 
 Let $z = re^{i\theta}$. Consider $z^n$.
 
+Since $z = r(\cos\theta + i\sin\theta)$,
 \begin{align*}
-\text{Since } z = r(\cos\theta + i\sin\theta) \\
 z^n &= r^n(\cos\theta + i\sin\theta)^n &\text{(1)} \\
-\text{But also} \\
+\end{align*}
+
+But also
+
+\begin{align*}
 z^n &= (re^{i\theta})^n \\
     &= r^n(e^{i\theta})^n \\
     &= r^ne^{in\theta} \\
     &= r^n(\cos{n\theta} + i\sin{n\theta}) &\text{(2)} \\
-\text{By equating (1) and (2), we find:}\\
+\end{align*}
+
+By equating (1) and (2), we find de Moivre's theorem:
+
+\begin{align*}
+r^n(\cos\theta +i\sin\theta)^n &= r^n(\cos{n\theta} + i\sin{n\theta}) \\
 (\cos\theta +i\sin\theta)^n &= (\cos{n\theta} + i\sin{n\theta})
 \end{align*}
 
+<details>
+<summary>
 ### Example 1
 
 Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$.
 
+</summary>
+
 > \begin{align*}
   r &= |1+i| = \sqrt2 \\
   \theta &= \arg{1+i} = \frac \pi 4 \\
@@ -249,7 +274,10 @@ Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$.
              &= 2^7 (1 - i) \\
              &= 128 - 128i
 > \end{align*}
+</details>
 
+<details>
+<summary>
 ### Example 2
 
 Use de Moivre's theorem to show that
@@ -260,6 +288,8 @@ Use de Moivre's theorem to show that
 \sin{2\theta} &= 2\sin\theta\cos\theta
 \end{align*}
 
+</summary>
+
 > Let $n=2$:
 
 > \begin{align*}
@@ -268,11 +298,17 @@ Use de Moivre's theorem to show that
   \text{Imaginary part: } 2\sin\theta\cos\theta &= \sin{2\theta}
 > \end{align*}
 
+</details>
+
+<details>
+<summary>
 ### Example 3
 
 Given that $n \in \mathbb{N}$ and $\omega = -1 + i$, show that
 $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula.
 
+</summary>
+
 > \begin{align*}
   r &= \sqrt{2} \\
   \arg \omega = \theta &= \frac 3 4 \pi \\
@@ -283,14 +319,20 @@ $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula.
                           &= 2^{\frac n 2 + 1}\cos{\frac {3n\pi} 4}
 > \end{align*}
 
+</details>
+
 ## Complex Roots of Polynomials
 
+<details>
+<summary>
 ### Example
 
-Which complex numbers $z$ satisfy
+Find which complex numbers $z$ satisfy
 
 $$z^3 = 8i$$
 
+</summary>
+
 > 1. Write $8i$ in exponential form,
 > 
 >    $|8i| = 8$ and $\arg{8i} = \frac \pi 2$
@@ -333,3 +375,5 @@ $$z^3 = 8i$$
 >        Some of these complex roots may be real numbers.
 > 
 > 4. There are three solutions
+
+</details>