DP Mathematics: Analysis and Approaches Questionbank

Consider quadrilateral $\mathrm{PQRS}$ where $\left[\mathrm{PQ}\right]$ is parallel to $\left[\mathrm{SR}\right]$.

In $\mathrm{PQRS}$, $\mathrm{PQ}=x$, $\mathrm{SR}=y$, $\mathrm{R}\hat{\mathrm{S}}\mathrm{P}=\alpha$ and $\mathrm{Q}\hat{\mathrm{R}}\mathrm{S}=\beta$.

Find an expression for $\mathrm{PS}$ in terms of $x, y, \sin \beta$ and $\sin \left(\alpha +\beta \right)$.

Show that $\text{arctan} p+\text{arctan} q\equiv \text{arctan}\left(\frac{p+q}{1-pq}\right)$ where $p, q>0$ and $pq<1$.

Verify that $\text{arctan\hspace{0.05em}}\left(2x+1\right)=\text{arctan\hspace{0.05em}}\left(\frac{x}{x+1}\right)\mathrm{+}\frac{\mathrm{\pi }}{4}$ for $x\in \mathrm{ℝ}, x>0$.

Verify that $x=\text{tan}\theta$ and $x=-\text{cot}\theta$ satisfy the equation $x^2+\left(2\text{cot}2\theta \right)x-1=0$.

Show that $\text{cot}2\theta =\frac{1-\text{ta}{\text{n}}^2\theta }{2\text{tan}\theta }$.

Hence, or otherwise, show that the exact value of $\text{tan}\frac{\pi }{12}=2-\sqrt{3}$.

Using the results from parts (b) and (c) find the exact value of $\text{tan}\frac{\pi }{24}-\text{cot}\frac{\pi }{24}$.

Give your answer in the form $a+b\sqrt{3}$ where $a$, $b\in \mathbb{Z}$.

Write down the amplitude of this graph

Find $\text{arctan}\frac{3}{4}$, giving the answer to 3 significant figures.

$A$.

Use your answers from part (a) to write down the value of $A$, $B$ and $D$.

Show that $\frac{b}{\sqrt{a^2+b^2}}=\text{cos}\theta$.

Hence prove your conjectures in part (e).

$C$.

Sketch the graph $y=3\text{sin}x+4\text{cos}x$, for $-2\pi ⩽x⩽2\pi$

Write down the period of this graph

Show that $\frac{a}{b}=\text{tan}\theta$.

Comment on your answer to part (c)(i).

Find the value of $C$.

$D$.

By considering the graph of $y=5\text{sin}x+12\text{cos}x$, find the value of $A$, $B$, $C$ and $D$.

$B$.

Solve $2\sin (x+{60}^{\circ })=\cos (x+{30}^{\circ }),0^{\circ }⩽x⩽{180}^{\circ }$.

Show that $\sin {105}^{\circ }+\cos {105}^{\circ }=\frac{1}{\sqrt{2}}$.

Find the modulus and argument of $z$ in terms of $\theta$. Express each answer in its simplest form.

Hence find the cube roots of $z$ in modulus-argument form.

Show that ${\left(\text{sin}x+\text{cos}x\right)}^2=1+\text{sin}2x$.

Show that $\text{sec}2x+\text{tan}2x=\frac{\text{cos}x+\text{sin}x}{\text{cos}x-\text{sin}x}$.

Hence or otherwise find ${\int }_0^{\frac{\pi }{6}}\left(\text{sec}2x+\text{tan}2x\right)\text{d}x$ in the form $\text{ln}\left(a+\sqrt{b}\right)$ where $a$, $b\in \mathbb{Z}$.

Let $f\left(x\right)=\text{tan}\left(x+\pi \right)\text{cos}\left(x-\frac{\pi }{2}\right)$ where .

Express $f\left(x\right)$ in terms of sin $x$ and cos $x$.

Find the roots of $z^{24}=1$ which satisfy the condition , expressing your answers in the form $r{e}^{\text{i}\theta }$, where $r$, $\theta \in {\mathbb{R}}^{+}$.

Show that Re S = Im S.

By writing $\frac{\pi }{12}$ as $\left(\frac{\pi }{4}-\frac{\pi }{6}\right)$, find the value of cos $\frac{\pi }{12}$ in the form $\frac{\sqrt{a}+\sqrt{b}}{c}$, where $a$, $b$ and $c$ are integers to be determined.

Hence, or otherwise, show that S = $\frac{1}{2}\left(1+\sqrt{2}\right)\left(1+\sqrt{3}\right)\left(1+\text{i}\right)$.

Find an expression for ${\theta }_p$ in terms of $p$.

Show that $p=\frac{q^2+q-1}{2q+1}$.

By sketching the graph of $p$ as a function of $q$, determine the range of values of $p$ for which there are possible values of $q$.

Find the value of $\sin \frac{\pi }{4}+\sin \frac{3\pi }{4}+\sin \frac{5\pi }{4}+\sin \frac{7\pi }{4}+\sin \frac{9\pi }{4}$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

$A$ and $B$ are acute angles such that $\text{cos}A=\frac{2}{3}$ and $\text{sin}B=\frac{1}{3}$.

Show that $\text{cos}\left(2A+B\right)=-\frac{2\sqrt{2}}{27}-\frac{4\sqrt{5}}{27}$.