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

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.

A landscape architect is designing a triangular garden bed $PQR$. The side $PQ$ is fixed at a length of $60$ metres. The angle $Q\hat{P}R$ is set to $35^\circ$. The architect initially plans for the side $PR$ to be $45$ metres.

(a) Calculate the length of the side $QR$.

Due to a change in design, the architect now wants the angle $P\hat{Q}R$ to be $75^\circ$. The side $PQ$ remains $60$ metres and $Q\hat{P}R$ remains $35^\circ$.

(b) Calculate the length of the side $PR$.

(c) Calculate the area of the garden bed $PQR$ with the dimensions from part (b).

The architect considers a different design where $PQ = 60$ m and $Q\hat{P}R = 35^\circ$ (as before). However, the side $QR$ is now fixed at $40$ metres.

(d) Show that two different triangular garden beds are possible, and find the two possible lengths for the side $PR$.

Find, in the form $re^{i\theta}$, the fourth roots of $-8 - 8\sqrt{3}i$.

The four roots from part (a) form the vertices of a quadrilateral in an Argand diagram. Determine the exact area of this quadrilateral.

Consider the non-zero vectors $\vec{u}$ and $\vec{v}$. Let $\theta$ be the angle between $\vec{u}$ and $\vec{v}$.

Using the definitions of $\vec{u} \cdot \vec{v}$ and $\vec{u} \times \vec{v}$ in terms of $|\vec{u}|$, $|\vec{v}|$ and $\theta$, show that $(\vec{u} \cdot \vec{v})^2 + |\vec{u} \times \vec{v}|^2 = |\vec{u}|^2|\vec{v}|^2$.

A triangle PQR has vertices P(1, 0, 1), Q($a, b$, 2) and R(4, 1, 1), where $a, b \in \mathbb{Q}$.

The vectors $\vec{u}$ and $\vec{v}$ are defined as $\vec{u} = \vec{PQ}$ and $\vec{v} = \vec{PR}$.

It is given that $\vec{u} \cdot \vec{v} = 4$ and the area of triangle PQR is $\frac{\sqrt{14}}{2}$ square units.

Find the value of $|\vec{u} \times \vec{v}|$.

Hence, or otherwise, find the value of $|\vec{u}|$.

Hence, or otherwise, find the possible values of $a$ and the corresponding values of $b$.

Consider a new point S, the vector $\vec{w}$ is defined as $\vec{w} = \vec{RS}$.

It is given that $\vec{u} \cdot \vec{w} = 0$ and $\vec{v} \cdot \vec{w} = 0$, and the area of triangle PRS is 10 square units.

Assuming that $a = 2$, find the possible vectors for $\vec{w}$.

A parallelogram has adjacent side lengths of 6 cm and 8 cm. Let $\alpha$ be the angle between these two sides. The area of the parallelogram is $12\sqrt{7}$ cm$^2$.

(a) Show that $\sin \alpha = \frac{\sqrt{7}}{4}$.

(b) Find the exact lengths of the two diagonals of the parallelogram.

(a) A surveyor is mapping a new development. From point A, the distance to a prominent tree (T) is measured as 300 m. From another point B, 250 m away from A, the distance to the same tree (T) is measured as 200 m.

Find the measure of angle ATB.

(b) Find the shortest distance from the tree (T) to the line segment AB.

A geological survey is being conducted in a mountainous region. Three sensor stations, A, B, and C, are set up at different locations. Their coordinates, relative to a central reference point (in meters), are given as:

Station A: $(1, 2, 3)$

Station B: $(4, 5, 6)$

Station C: $(7, 2, 1)$

(a) Find the distance between Station A and Station B.

(b) Find the size of the angle $\text{ABC}$ (the angle at Station B).

A landscape architect is designing a triangular shade sail for a patio. The vertices of the sail are defined by points A, B, and C in a 3D coordinate system, where the z-axis represents height.

The coordinates of the vertices are A$(0, k, 1)$, B$(2, 1, 0)$ and C$(k, 0, 3)$, where $k$ is a positive constant representing a design parameter.

(a) Show that the vector product $\vec{AB} \times \vec{AC}$ is given by $\begin{pmatrix} 2-3k \\ -k-4 \\ k^2-3k \end{pmatrix}$.

(b) The architect wants to minimize the tension in the sail, which is proportional to the magnitude of the vector product of two adjacent sides. Find the smallest possible value of $|\vec{AB} \times \vec{AC}|$.

(c) Calculate the smallest possible area of the shade sail.

A plan for a garden is drawn on a coordinate grid, where 1 unit represents 1 metre. The garden consists of a paved patio and a flower bed.

The patio is a quadrilateral with vertices A(-4, 2), B(0, 5), C(4, 2), and D(0, -1).

(a) Find the area of the patio ABCD.

The flower bed is a triangle with vertices C(4, 2), E(9, 4), and F(9, 0).

(b) Find the area of the flower bed CEF.

(c) Hence, find the total area of the garden.

(a) Calculate the shortest distance from the apex $V$ to the side $AB$ of the pyramid's base.

(b) Find the total surface area of the four triangular glass panels forming the pyramid's roof.

(c) Calculate the total height of the monument from the ground to the apex $V$.

(d) Determine the size of the angle between the slant edge $VA$ and the base edge $AB$.

(e) A security camera is installed at a point $S$ on the vertical edge $C'C$ of the monument. A guard standing at point $B'$ (a corner of the base of the monument) measures the angle of elevation to point $S$ to be $40^\circ$. Find the height of the security camera from the ground.

(a) A landscape architect is designing a new park. From a central monument, two paths, Path A and Path B, diverge. Path A is $75$ m long, and Path B is $60$ m long. The angle between Path A and Path B at the monument is $125^\circ$.

Calculate the direct distance between the ends of Path A and Path B.

(b) Calculate the area of the triangular region enclosed by Path A, Path B, and the direct line connecting their ends.

(c) Calculate the angle that Path B makes with the direct line connecting the ends of the paths.