Show that $$1 + \mathrm{e}^{i2\theta} = 2\cos\theta(\cos\theta + i\sin\theta)$$
Given $$\mathrm{C} = 1 + \begin{pmatrix}n\\1\end{pmatrix}\cos 2\theta + \begin{pmatrix}n\\2\end{pmatrix}\cos 4\theta + ... + \cos 2n\theta$$ $$\mathrm{S} = \begin{pmatrix}n\\1\end{pmatrix}\sin 2\theta + \begin{pmatrix}n\\1\end{pmatrix}\sin 4\theta + ... + \sin 2n\theta$$ and by considering $\mathrm{C} + i\mathrm{S}$, find an expression for $\mathrm{S}$.
Factor out $\mathrm{e}^{i\theta}$
$2^n\cos^n\theta\sin n\theta$
Q2
Answer
Find the general solution of the differential equation $$\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3$$
$y = A + B\mathrm{e}^{2x} - \dfrac{3}{2}x$
Q3
Answer
With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a} = \mathbf{i} + 2\mathbf{j}$, $\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}$ and $\mathbf{c} = 3\mathbf{i} - \mathbf{j} + \mathbf{k}$.
Two lines $L_1$ and $L_2$ have vector equations $\mathbf{r} = \begin{pmatrix}1\\2\\3\end{pmatrix} + \lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$ and $\mathbf{r} = \begin{pmatrix}3\\0\\1\end{pmatrix} + \mu\begin{pmatrix}1\\-1\\2\end{pmatrix}$.
Show that $L_1$ and $L_2$ intersect and find the point of intersection.
Find the acute angle between $L_1$ and $L_2$.
(a) Intersection at $(3, 3, 2)$ $\quad$ (b) $\approx 79.1°$