HL Circles (Radians): notes and practice questions

(a) A landscape architect is designing a triangular garden bed. Two sides of the bed measure $8$ m and $15$ m, respectively. The angle between these two sides is $\frac{3\pi}{4}$ radians.

Calculate the area of the garden bed.

(b) Calculate the length of the third side of the garden bed.

(a) A ground crew on Luna Prime is tracking a supply drone. From the Aether station's control tower, the drone is observed to be $8500$ km East and $4000$ km North in a localized flat-map approximation.

(i) Find the straight-line distance from the Aether station to the drone.

(ii) Find the bearing of the drone from the Aether station. Give your answer in degrees, correct to one decimal place.

(b) The Aether station (A) is located at the origin $(0,0,0)$ of a 3D Cartesian coordinate system for this part. A research probe (P) is located at $(4000, 0, 0)$ km. A navigation beacon (B) is located at $(0, 4000, 0)$ km.

(i) Show that the position vector of the research probe, $\vec{p}$, is perpendicular to the position vector of the navigation beacon, $\vec{b}$.

(ii) Assuming the probe and beacon are on the surface of Luna Prime with a radius of $4000$ km, calculate the shortest distance along the surface between the research probe and the navigation beacon.

(c) Consider the Aether station (A) at $(4000, 0, 0)$ km, the Boreas station (B) at $(0, 4000, 0)$ km, and a North Pole reference point (N) at $(0, 0, 4000)$ km. Let $\vec{a}$, $\vec{b}$, and $\vec{n}$ be their respective position vectors from the centre of Luna Prime.

(i) Find the vector $\vec{a} \times \vec{b}$.

(ii) Show that the angle at vertex A in the spherical triangle formed by Aether, Boreas, and the North Pole (ABN) is $90^{\circ}$.

(d) A supply route between Aether and a new outpost, Delta, has an arc length of $2000$ km. Given that the radius of Luna Prime is $4000$ km, show that the central angle $\theta$ between Aether and Delta is $28.6^{\circ}$, correct to three significant figures.

(e) The Aether station (A) is located at $40^{\circ}$ N, $20^{\circ}$ E, and the Boreas station (B) is located at $70^{\circ}$ N, $100^{\circ}$ E on Luna Prime, which has a radius of $6000$ km. Find the shortest distance along the surface from Aether to Boreas.

(f) Using the vector method from part (c), find the initial bearing from Aether to Boreas. Give your answer in degrees, correct to one decimal place.

(a) A weather satellite is in a circular orbit around a planet. Its velocity vector, relative to a fixed ground station, is given by

$\mathbf{v} = \begin{pmatrix} -100\pi \cos(2\pi t) \\ -100\pi \sin(2\pi t) \end{pmatrix}$ m/s

where $t$ is the time in seconds. Distances are measured in metres.

Find its acceleration vector $\mathbf{a} = \begin{pmatrix} \ddot{x}(t) \\ \ddot{y}(t) \end{pmatrix}$.

(b) At time $t=0$ seconds, the satellite is at the position $(0, 150)$ m. Find its displacement vector $\mathbf{s} = \begin{pmatrix} x \\ y \end{pmatrix}$.

(c) When the vertical acceleration of the satellite is zero, find the possible positions for the satellite.

(d) State how many complete orbits the satellite makes in one minute.

In a triangular plot of land XYZ, the side XY measures $5\text{ m}$, the side YZ measures $4\text{ m}$, and the angle $Y\hat{X}Z$ is $\frac{\pi}{6}$ radians.

Use the cosine rule to find the two possible lengths for the side XZ.

Find the absolute difference between the areas of the two possible triangular plots of land XYZ.