Non-right angle triangles (Sine, cosine rule, area of triangle): notes and practice questions

The diagram below shows a hot air balloon hovering at point H, $520$m vertically above a landing pad.

Point A is the point on the ground, directly below the hot air balloon.

An observer starts walking at a constant speed from point C towards point A. From point C, the observer looks upward at the hot air balloon at an angle of elevation of $30\degree$. After $20$ minutes, the observer is at point B and observes the same hot air balloon at an angle of elevation of $50\degree$.

Write down the size of the angle of depression from H to C.

Find the horizontal distance from A to C.

Calculate the distance the observer walked from C to B.

Determine the observer's average speed, in metres per hour.

A drone is used for aerial surveying. It starts at a central hub (H).

(a)(i) From H, it flies 15 km on a bearing of 040° to survey point S1. From S1, it flies 10 km on a bearing of 170° to survey point S2.

Determine the interior angle HS1S2.

(a)(ii) Determine the distance from S2 back to H.

(a)(iii) Determine the bearing the drone must use to travel directly from S2 to H.

(b) The drone's mission also includes a final delivery point D, such that S1S2DH forms a parallelogram. Write down the distance between point D and point H. Justify your answer.

Three remote research stations, Alpha (A), Beta (B), and Gamma (C), are located in a vast desert region. The distances between these stations are given below:

• Distance between Alpha and Beta (AB) = 250 km

• Distance between Beta and Gamma (BC) = 320 km

• Distance between Gamma and Alpha (CA) = 280 km

(a) Calculate the measure of angle ABC, in degrees, correct to one decimal place.

(b) Find the area of the triangular region formed by the three research stations, in $km^2$, correct to three significant figures.

A modern architectural marvel, 'The Apex Tower', is designed with a square base and a single peak directly above the centre of the base. The side length of the square base is $240 \text{ m}$, and the vertical height of the tower from the centre of the base to its peak is $160 \text{ m}$.

Let the base be represented by square ABCD and the peak by V. Let M be the midpoint of the side AB.

(a) Calculate all the side lengths and angles of the triangle $\triangle VOM$, where O is the centre of the square base.

(b) Calculate all the side lengths and angles of the triangle $\triangle VAB$.

A graphic designer is creating a logo for a new technology company. The logo features a stylized 'C' shape, which is a circular arc. This arc is part of a circle with centre O and a radius of 8 cm. The arc subtends an angle of 120° at the centre O. The straight edge of the 'C' is a chord connecting the endpoints of the arc.

The logo component is shown as the curved boundary in the following diagram.

(a) Find the length of the curved boundary of this logo component.

(b) Find the area of the region enclosed by the arc and the straight edge (the segment of the circle).

A rectangular prism with length 10 cm, width 8 cm, and height 6 cm is used as a base for a sculpture. One corner, vertex H, is cut off by a flat plane passing through points P, Q, and R. Point P is on edge HE such that HP = 3 cm. Point Q is on edge HG such that HQ = 4 cm. Point R is on edge HD such that HR = 3 cm.

(a) Calculate the volume of the remaining part of the prism.

(b) Calculate the total surface area of the remaining part of the prism.

A triangular section of a modern architectural roof is being designed. The vertices of this section are represented by points P, Q, and R in a 3D coordinate system, where distances are measured in metres.

Point P is at $(0, 0, 2)$.

Point Q is at $(4, 7, 3)$.

Point R is at $(9, 2, 2)$.

Calculate the vector product $\vec{PQ} \times \vec{PR}$.

Hence, find the area of the triangular roof section.

A support beam is to be installed from point Q perpendicular to the edge PR. Find the length of this support beam.

The building's base lies on the horizontal plane (z=0). Find the acute angle that the roof section makes with the horizontal plane.

A specialized navigation chart uses complex numbers to represent locations. The central hub is at the origin $(0,0)$. Two important landmarks, 'Anchor Point' ($z_A$) and 'Beacon Tower' ($z_B$), are located at $z_A = 5 - 2i$ and $z_B = 4e^{i\frac{\pi}{3}}$ respectively.

(a) (i) Find the modulus of $z_A$.

(a) (ii) Find the argument of $z_A$, giving your answer in radians.

(b) Find the area of the triangular region formed by the central hub, Anchor Point ($z_A$), and Beacon Tower ($z_B$).

The matrices $P$ and $Q$ are defined by $P = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $Q = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$.

(a) Describe fully the single geometrical transformation represented by $P$.

A quadrilateral $K$ is mapped onto quadrilateral $L$ by the composite transformation represented by the matrix product $PQ$. The coordinates of the vertices of $L$ are (0, 0), (0, 2), (−1, 1) and (−1, 3).

(b) Find the coordinates of the vertices of $K$.

(c) (i) Find the area of quadrilateral $K$.

(c) (ii) Hence, find the area of quadrilateral $L$.

The matrix $M = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ represents a combination of transformations:

A rotation of 90° clockwise about the origin;

Followed by a horizontal stretch with scale factor 2, with the y-axis invariant;

Followed by a transformation represented by matrix $F$.

(d) Find matrix $F$.

A surveyor is mapping a triangular plot of land, ABC. The length of side AB is 12.0 km, the length of side BC is 20.0 km, and angle A is 35 degrees.

(a) Find the length of side AC using one application of the cosine rule.

(b) Find the length of side AC using two applications of the sine rule.

A triangular sail for a yacht has side lengths of $12$ m, $15$ m, and $20$ m.

Find the size of the largest interior angle of the sail, correct to one decimal place.

Hence find the area of the sail, correct to three significant figures.

A triangular sail, $\triangle PQR$, is being designed for a yacht. The angle at vertex P, $Q\hat{P}R$, measures $72^\circ$. The side $QR$ has a length of $12.5$ m and the side $PQ$ has a length of $10.8$ m.

Calculate the measure of angle $P\hat{R}Q$.

Hence find the area of $\triangle PQR$.

A high-altitude drone takes off from a research station, point P. The drone flies due east at a constant speed of 50 km/h. It departs at 08:00. At 10:00, it reaches point Q and changes its course. It then flies for another 1.5 hours at the same speed, reaching point R. The course adjustment at Q is 40° southward from its original eastward direction.

(a) Determine the distance of the drone from the research station (PR) at 11:30. Write down your answer correct to the nearest integer.

(b) Determine the angle R$\hat{P}$Q. Give your answer correct to one decimal place.

Three observation points in a 3D geological survey are located at points P, Q, and R. Their position vectors relative to an origin O are given by $\vec{OP} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$, $\vec{OQ} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}$ and $\vec{OR} = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}$.

(a) Calculate $\vec{PQ} \times \vec{PR}$.

(b) Hence find the area of triangle PQR.

(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 surveyor is mapping a triangular plot of land. The lengths of the three sides of the plot are measured as $50$ m, $60$ m, and $70$ m.

(a) Calculate the measure of the largest interior angle of the plot. Give your answer in degrees, correct to one decimal place.

(b) Find the area of the triangular plot of land. Give your answer correct to three significant figures.

A designer is creating a decorative arch for a garden entrance. The cross-section of the arch can be modelled by the function $f(x) = x(6-x)$, for $x \in \mathbb{R}$, where $x$ and $f(x)$ are measured in metres. The arch is supported by the ground (the $x$-axis) and starts at the origin (the $y$-axis).

(a) Write down an integral expression for the area of the cross-section of the arch.

(b) Find the area of the cross-section of the arch.

To stabilize the arch, a triangular support panel is to be fitted. The vertices of this triangular panel are at $P(0, 0)$, $Q(4, 9)$, and $R(k, 0)$. The area of this triangular panel must be equal to the area of the arch's cross-section calculated in part (b).

(c) Find the value of $k$, the $x$-coordinate of $R$.

A circular park has a central monument. A straight walking path $AB$ cuts across the park. The radius of the park is $6$ m. The shortest distance from the center of the park $O$ to the path $AB$ is $5$ m.

(a) (i) Calculate the angle $A\hat{O}B$ in degrees.

(a) (ii) The region of the park bounded by the path $AB$ and the arc $AB$ is a flower bed. Calculate the area of this flower bed.

A square performance stage has a special effect light at its center. The stage has a side length of $10$ m. The light projects a circular pattern on the stage floor with a radius of $6$ m.

(b) (i) Calculate the total area of the circular light pattern if it were projected onto an infinite surface.

(b) (ii) Calculate the area of the stage floor that is lit by the special effect light.

Let $B$ be the brightness of the special effect light, measured in lumens, and $t$ be the time in minutes since the light was switched on.

The rate of change of brightness is given by $\frac{dB}{dt} = 0.5 t e^{-0.2t}$.

(c) Find the value of $t$ at which the brightness of the light is increasing at the greatest rate.

A drone is used for aerial surveillance of a research facility. Three key locations are the Launch Pad (L), a Data Collection Point (D), and a Scientist's observation post (S).

The distance between the Launch Pad (L) and the Data Collection Point (D) is $LD = 10.0$ m.

The Scientist (S) is observing such that $L\hat{S}D = 30^\circ$. The distance from the Scientist to the Launch Pad is $SL = 10.0$ m. The angle $S\hat{L}D$ is obtuse.

Calculate the size of $S\hat{D}L$.

Calculate the area of triangle $SDL$.

An Automated Repair Station (R) is located such that it is $LR = 8.0$ m from the Launch Pad (L). The angle $D\hat{L}R = 60^\circ$.

Calculate the distance between the Data Collection Point (D) and the Repair Station (R).

A new Target (T) is identified. The distance from the Repair Station (R) to the Target (T) is $RT = 22.2$ m. From the Scientist's observation post (S), the angle $R\hat{S}T = 53.8^\circ$. From the Repair Station (R), the angle $S\hat{R}T = 51.1^\circ$.

Determine whether the Scientist (S) or the Repair Station (R) is closer to the Target (T).

The drone needs to travel from the Repair Station (R) to the Target (T) along a semi-circular maintenance track. This track has its centre at the Launch Pad (L). The angle subtended by the arc $RT$ at the centre L is $110^\circ$.

Calculate the distance the drone travels along this track.