Unit circle, exact values of main trig ratios, ambiguous case: notes and practice questions

(a) Show that $2x+5 + \frac{3}{x-1} = \frac{2x^2 + 3x - 2}{x-1}$, for $x \in \mathbb{R}, x \neq 1$.

(b) Hence or otherwise, solve the equation $2\sin{\theta} + 5 + \frac{3}{\sin{\theta}-1} = 0$ for $0 \leq \theta \leq 2\pi$, $\theta \neq \frac{\pi}{2}$.

Consider the complex numbers $w_1 = 1 + k\text{i}$ and $w_2 = k - \text{i}$, where $k \in \mathbb{R}$ and $k \neq 0$.

(a) Find an expression for $w_1 w_2$ in terms of $k$.

(b) Hence, given that $\text{arg}(w_1 w_2) = \frac{\pi}{6}$, find all possible values of $k$.

Solve the following equations for $\theta$ in the interval $0 \le \theta \le 2\pi$. Give your answers as multiples of $\pi$.

(a) $\cos \theta = -\frac{\sqrt{3}}{2}$

(b) $\sin \theta = -\frac{1}{2}$

(c) $\tan \theta = -\sqrt{3}$

A ladder must be placed against a tall vertical building, clearing a monument that is 8 m high and stands on horizontal ground 1 m away from the building's base. The ladder touches the ground, the top corner of the monument, and the wall of the building.

Let $L$ be the length of the ladder in metres.

Let $\theta$ be the angle that the ladder makes with the ground, where $0 < \theta < \frac{\pi}{2}$.

(a) Show that $L = \sec \theta + 8\csc \theta$.

(b) (i) Find $\frac{dL}{d\theta}$.

(b) (ii) When $\frac{dL}{d\theta} = 0$, show that $\tan\theta = 2$.

(c) (i) Find $\frac{d^2L}{d\theta^2}$.

(c) (ii) When $\tan\theta = 2$, find the value of $\frac{d^2L}{d\theta^2}$.

(d) (i) Hence, justify that $L$ is a minimum when $\tan\theta = 2$.

(d) (ii) Determine this minimum value of $L$.

(e) A construction company only has ladders with a maximum length of 11 m. Determine whether it is possible to position a ladder against the building over the monument, giving a reason for your answer.

Consider the functions $f(x) = \sin x - \cos x$ and $h(x) = x + \frac{\pi}{4}$ for $x \in \mathbb{R}$.

(a) Find an expression for $(f \circ h)(x)$.

(b) Hence, solve the equation $(f \circ h)(x) = 1$ for $0 \le x \le 2\pi$.

By using an appropriate substitution, show that $\int \sin(\sqrt{x}) \, dx = 2\sin(\sqrt{x}) - 2\sqrt{x} \cos(\sqrt{x}) + C$.

The following diagram shows part of the curve $y = \sin(\sqrt{x})$ for $x \ge 0$.

The curve intersects the x-axis at $x_1, x_2, x_3, \dots$.

The nth x-intercept of the curve, $x_n$, is given by $x_n = n^2 \pi^2$, where $n \in \mathbb{Z}^+$.

Write down an expression for $x_{n+1}$.

The regions bounded by the curve and the x-axis are denoted by $R_1, R_2, R_3, \dots$ as shown on the diagram.

Calculate the area of region $R_n$.

Give your answer in the form $(an+b)\pi$, where $a, b \in \mathbb{Z}^+$.

Hence, show that the areas of the regions $R_1, R_2, R_3, \dots$ form an arithmetic sequence.

Show that the equation $2\sin^2x - \cos x - 1 = 0$ may be written in the form

$2\cos^2x + \cos x - 1 = 0$.

Hence, solve the equation $2\sin^2x - \cos x - 1 = 0$, for $0 \le x \le 2\pi$.

Consider the functions $f(x) = -2\cos(x) + 5$ and $g(x) = -2\cos(x - \frac{\pi}{4}) + 5 - k$, where $x \in \mathbb{R}$ and $k > 0$.

The graph of $g$ is obtained by two transformations of the graph of $f$.

(a) Describe these two transformations.

The $y$-intercept of the graph of $g$ is at $(0, s)$.

(b) Given that the maximum value of $g(x)$ is 1, find the value of $s$.

Show that $3x + 2 - \frac{2}{x+1} = \frac{3x^2 + 5x}{x+1}$, for $x \in \mathbb{R}, x \neq -1$.

Hence or otherwise, solve the equation $3\cos(\theta) + 2 - \frac{2}{\cos(\theta)+1} = 0$ for $0 \le \theta \le 2\pi, \theta \neq \pi$.

Consider an acute angle $x$ such that $\cos x = \frac{3}{4}$.

Find the value of $\sin x$.

Find the value of $\cos(2x)$.

The following diagram shows triangle ABC, with $\text{B}\hat{\text{C}}\text{A} = x$, $\text{B}\hat{\text{A}}\text{C} = 2x$, and AB = 14.

(b) Show that BC = 21.

The line segment CA is extended to a point D such that triangle ABD is isosceles with AB = AD.

Find the size of angle ADB in terms of $x$.

Find the area of triangle ABD.

Consider the complex number $z_1 = 1 - i\sqrt{3}$.

(a) (i) Express $z_1$ in modulus-argument form.

(a) (ii) Find the smallest positive integer $n$ for which $z_1^n$ is a real number.

Consider the equation $(z+i)^5 - (z-i)^5 = 0$, where $z \in \mathbb{C}$.

(b) Show that the roots of the equation are given by $z = \cot\left(\frac{k\pi}{5}\right)$ for $k=1, 2, 3, 4$.

(c) (i) By using the binomial expansion, show that the equation in part (b) can be written as $5z^4 - 10z^2 + 1 = 0$.

(c) (ii) Let the roots of the equation in (c)(i) be $z_1, z_2, z_3, z_4$. Without finding the roots, show that $\sum_{j=1}^4 z_j^2 = 4$.

(d) Hence, find the exact value of $\cot^2\left(\frac{\pi}{5}\right)$.

Consider the functions $f(x) = \sin x + \sqrt{3}\cos x$ and $g(x) = 2x$.

(a) Find $(f \circ g)(x)$.

(b) Solve the equation $(f \circ g)(x) = 2\sin(2x)$ for $0 \leq x \leq \pi$.

A team of engineers is testing two autonomous robots, Alpha and Beta, on a straight track. Their positions are measured as the distance from a fixed starting point. The experiment runs for 10 minutes.

The position of Robot Alpha, $P_A$ metres, at time $t$ minutes can be modelled by the function $P_A(t) = 3\sin(2t + 5) + 14t + 20$, where $0 \le t \le 10$.

The position of Robot Beta, $P_B$ metres, at time $t$ minutes can be modelled by the function $P_B(t) = 12t + 25$, where $0 \le t \le 10$.

Use the engineers' models to find the initial position of

(i) Robot Beta;

(ii) Robot Alpha correct to three significant figures.

Find the values of $t$ when Robot Alpha and Robot Beta are at the same position. Give your answers correct to three significant figures.

For $t > 5$, prove that Robot Alpha was always ahead of Robot Beta.

For $0 \le t \le 10$, find the total amount of time when the speed of Robot Beta was greater than the speed of Robot Alpha. Give your answer correct to three significant figures.

Show that the equation $\cos(2x) + \cos(x) = 0$ can be written in the form $2\cos^2(x) + \cos(x) - 1 = 0$.

Hence, solve $\cos(2x) + \cos(x) = 0$, for $0 \le x \le 2\pi$.

Find $\int x \sec^2 x \, dx$.

The region $R$ is enclosed by the curve $y = x \sec^2 x$, the $x$-axis, and the lines $x=0$ and $x=\frac{\pi}{3}$. Show that the area of $R$ is $\frac{\pi\sqrt{3}}{3} - \ln 2$.

Consider the function $g(x) = \csc\left(x+\frac{\pi}{3}\right)$, for $0\le x\le \frac{\pi}{3}$.

Determine the range of $g$.

The region bounded by the graph of $y = g(x)$, the $x$-axis and the lines $x = 0$ and $x = \frac{\pi}{3}$ is rotated $2\pi$ radians about the $x$-axis.

Find the volume of the solid generated.

Given that $\cos \theta = \frac{5}{13}$, and $\frac{3\pi}{2} < \theta < 2\pi$,

find $\sin \theta$ and $\tan \theta$.

Solve $3 \cos x = \cot x$ for $0 \le x \le 2\pi$.