Unit circle, pythagorean identity, tan(x), ambiguous case: notes and practice questions

A large observation wheel, 'The SkyGazer', rotates at a constant speed. The height, $h$ metres, of a passenger capsule above the ground can be modelled by the function $h(t) = a \sin(bt) + c$, where $t$ is the time in minutes since the capsule started its ascent from the wheel's horizontal midline. The ride starts at 10:00 AM.

At 10:00 AM, the capsule is at its midline height and rising. It reaches its maximum height of 65 m at 10:04 AM. Its lowest point is 5 m above the ground.

(a) Find the value of $a$.

(b) Find the value of $b$.

(c) Find the first time after the ride starts when the capsule reaches a height of 60 m. Give your answer to the nearest minute.

(a) A drone launches a rescue package from an initial position of $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ metres, relative to an origin on the ground. The package is launched with an initial speed of $15\text{ms}^{-1}$ at an angle $\theta$ to the horizontal ground, where $0 < \theta < \frac{\pi}{2}$.

The velocity components of the package, $t$ seconds after it is launched, are given by $v_x(t) = 15\cos\theta$ and $v_y(t) = 15\sin\theta - 9.8t$.

Find an expression for $x$, the horizontal displacement from the origin, in terms of $\theta$ and $t$.

(b) It is given that the vertical displacement of the package from the ground is $y = (15\sin\theta)t - 4.9t^2$. When the package hits the ground, show that $t = \frac{150\sin\theta}{49}$.

(c) Let $x_g$ be the value of $x$ when the package hits the ground.

Find an expression for $x_g$ in terms of $\theta$ only.

(d) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

(e) The model is adapted to account for a horizontal wind with speed $2\text{ms}^{-1}$ acting in the opposite direction to the initial horizontal motion.

(i) In this new model, the horizontal velocity component is $v_x(t) = 15\cos\theta - 2$. The time taken for the package to hit the ground remains $t = \frac{150\sin\theta}{49}$.

Find an expression for $x_g$, the value of $x$ when the package hits the ground, in terms of $\theta$ only.

(ii) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

A drone's camera gimbal can rotate to adjust its view. The transformation matrix $R(\theta)$ that rotates a point $(x, y)$ on the camera's image plane counter-clockwise about the origin through an angle $\theta$ is given by:

$R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$

The drone needs to perform a total rotation of $2\theta$ for a panoramic shot. Write down the matrix $R(2\theta)$ that represents this single combined rotation.

Alternatively, the drone's gimbal controller performs two consecutive rotations, each through an angle of $\theta$. Calculate the resulting transformation matrix when $R(\theta)$ is applied twice, i.e., $R(\theta) \times R(\theta)$.

By comparing your results from part (a) and part (b), explain how the identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ can be derived.

Using the same comparison as in part (c.i), and the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, show that $\cos(2\theta) = 1 - 2\sin^2(\theta)$.

A complex number is defined by $z = \sin(2\theta) - \text{i}(1 + \cos(2\theta))$ for $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.

(a) Solve $3 \cos(x - 45^{\circ}) = \sin(x + 45^{\circ})$ for $0^{\circ} \le x \le 360^{\circ}$.

(b) Show that $\cos 15^{\circ} - \sin 15^{\circ} = \frac{1}{\sqrt{2}}$.

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

(d) Hence find the fourth roots of $z$ in modulus-argument form.

A design studio is manufacturing a decorative wall sconce. The cross-sectional profile of the sconce is defined by two mathematical curves. The sconce is formed by rotating the region between these curves through $\pi$ radians about the $x$-axis, creating a flat rear surface that mounts flush against a wall. All linear dimensions are in centimetres.

The curve of the outer profile is given by $f(x) = 6 \cos\left(\frac{\pi}{12}x\right)$, for $0 \le x \le 6$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.6$ units to the left and $1.2$ units down.

(a) Write down an expression for $g(x)$.

(b) The curve $g(x)$ intersects the $x$-axis at $(a, 0)$, where $0 \le x \le a$ defines the inner boundary of the sconce.

Find the value of $a$. Give your answer to three significant figures.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the sconce. Give your answer to three significant figures.

A design studio is developing a wall-mounted decorative bracket. The bracket is designed to mount flush against a flat vertical wall. The cross-sectional profile of the bracket in the $x y$-plane is defined by two mathematical curves, where all linear dimensions are measured in centimetres.

The curve of the outer profile is given by $f(x) = 8 \cos\left(\frac{\pi}{16}x\right)$, for $0 \le x \le 8$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.8$ units to the left and $1.2$ units downwards.

(a) Write down an expression for $g(x)$.

The inner profile curve $g(x)$ intersects the $x$-axis at the point $(a, 0)$. The relevant portion of the inner profile is restricted to $0 \le x \le a$.

(b) Find the value of $a$. Give your answer to three significant figures.

The bracket is modelled by the solid formed when the region between the profiles is rotated through $\pi$ radians about the $x$-axis. This region is bounded by $f(x)$ from $x = 0$ to $x = 8$, the line $x = 0$, the $x$-axis, and $g(x)$ from $x = 0$ to $x = a$.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the bracket. Give your answer to three significant figures.

The height of the tide in a harbour, $H$ metres, $t$ hours after midnight, can be modelled by the function $H(t) = 3.5 + 1.8 \cos \left( \frac{\pi t}{6} \right)$.

(a) Find the minimum and maximum heights of the tide.

(b) Determine the first two positive values of $t$ for which the tide height is $2.6$ metres.

(a) The temperature $T$ (in degrees Celsius) inside a specialized plant incubator varies with time $t$ (in minutes) according to the model $T = 25.0 - 1.5 \cos\left(\frac{\pi}{30}t\right)$.

Determine the maximum and minimum temperatures inside the incubator.

(b) Find the first time after $10$ hours at which the temperature is $24.0 \, ^{\circ}\text{C}$. Give your answer to three significant figures.

A team of architects is designing a new domed stadium. The cross-section of the dome's roof can be modelled by the function $f(x) = 0.8e^{-0.5x^2}$, where $-3 < x < 3$. The $x$-axis represents the ground level and the $y$-axis represents the height above the ground, with units in metres.

A support beam is to be anchored to the dome's roof at a point, P, where the roof's height is $0.6$ m above the ground. The coordinates of P are $(a, 0.6)$, where $a > 0$.

Calculate the value of $a$.

Find an expression for $f'(x)$.

The support beam is designed to be perpendicular to the dome's roof at point P. Find the angle, $\theta$, that this support beam makes with the horizontal ground.

A company is designing a decorative stand for a new line of luxury smart speakers. The profile of the stand's base is defined by two mathematical curves. The stand itself is formed by rotating the region between these curves through $\pi$ radians about the $x$-axis. All dimensions are in centimetres.

The curve of the outer profile is given by $f(x) = 5 \cos\left(\frac{\pi}{10}x\right)$, for $0 \le x \le 5$.

The curve of the inner profile, $g(x)$, is formed by translating the graph of $f$ by $0.4$ units to the left and $0.6$ units down.

(a) Write down an expression for $g(x)$.

The inner profile curve $g(x)$ intersects the $x$-axis at $(a, 0)$. The relevant portion of $g(x)$ for the stand is restricted to $0 \le x \le a$.

(b) Find the value of $a$. Give your answer to three significant figures.

The decorative stand is modelled by the solid formed when the region $R$ is rotated through $\pi$ radians about the $x$-axis. The region $R$ is defined by the area under $f(x)$ from $x=0$ to $x=5$, excluding the area under $g(x)$ from $x=0$ to $x=a$.

(c.i) Write down an expression for the volume of the solid formed.

(c.ii) Hence find the volume of material used in the stand. Give your answer to three significant figures.