TRIGONOMETRY · HL

3D Diagrams

Pull the 2D triangles out of a 3D figure.

3D DIAGRAMS

Corner of a room

Section 1 of 3

Two vertical poles

$[sp]$, $[tq]$ are vertical poles each of height 10 m. $p$, $q$, $r$ are points on level ground. Two wires of equal length join $s$ and $t$ to $r$, i.e. $|sr| = |tr|$.

If $|pr| = 8$ m, $|\angle pqr| = 30^\circ$, $|\angle prq| = 120^\circ$, calculate

(i) $|pq|$ to the nearest metre

(ii) $|sr|$ in surd form

(iii) $|\angle srt|$ to the nearest degree.

(i) $|pq|$ — Sine Rule

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$

$\dfrac{a}{\sin 120^\circ} = \dfrac{8}{\sin 30^\circ}$

$a = \dfrac{8\,\sin 120^\circ}{\sin 30^\circ}$

$a = 13.9 = 14$ m

(ii) $|sr|$ — Pythagoras

$x^{2} = 8^{2} + 10^{2}$

$x^{2} = 164$

$x = \sqrt{164}$

(iii) $|\angle srt|$ — Cosine Rule

$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$

$14^{2} = 164 + 164 - 2(164)\cos A$

$196 = 328 - 328\cos A$

$328\cos A = 132$

$\cos A = \dfrac{33}{82}$

$A = 66^\circ$

Section 2 of 3

Pole with two angles of elevation

$p$, $q$ and $r$ are three points on horizontal ground. $[sr]$ is a vertical pole of height $h$ metres.

The angle of elevation of $s$ from $p$ is $60^\circ$ and the angle of elevation of $s$ from $q$ is $30^\circ$. $|pq| = c$ metres.

Given that $3c^{2} = 13h^{2}$, find $|\angle prq|$.

From $p$ (elevation $60^\circ$)

$\tan 60^\circ = \dfrac{h}{x}$

$\sqrt{3} = \dfrac{h}{x}$

$\sqrt{3}\,x = h$

$x = \dfrac{h}{\sqrt{3}}$

From $q$ (elevation $30^\circ$)

$\tan 30^\circ = \dfrac{h}{y}$

$\dfrac{1}{\sqrt{3}} = \dfrac{h}{y}$

$y = \sqrt{3}\,h$

$|\angle prq|$ — Cosine Rule

$\dfrac{2h}{\sqrt{3}} \cdot \dfrac{\sqrt{3}\,h}{1} = 2h^{2}$

$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$

$c^{2} = a^{2} + b^{2} - 2ab\,\cos C$

$c^{2} = \dfrac{h^{2}}{3} + 3h^{2} - 2\cdot\dfrac{h}{\sqrt{3}}\cdot\sqrt{3}\,h\,\cos C$

$3c^{2} = h^{2} + 9h^{2} - 6h^{2}\cos C$

$13h^{2} = 10h^{2} - 6h^{2}\cos C$

$6\cos C = -3$

$\cos C = -\dfrac{1}{2}$

$C = 120^\circ$

Section 3 of 3

The Great Pyramid at Giza

The great pyramid at Giza in Egypt has a square base and four triangular faces. The base of the pyramid is of side 230 metres and the pyramid is 146 metres high. The top of the pyramid is directly above the centre of the base.

(i) Calculate the length of one of the slanted edges, correct to the nearest metre.

(ii) Calculate, correct to two significant figures, the total area of the four triangular faces of the pyramid (assuming they are smooth flat surfaces).

Half-diagonal of the base

$t^{2} = 115^{2} + 115^{2}$

$t^{2} = 2(115)^{2}$

$t = 115\sqrt{2}$

(i) Slanted edge

$y^{2} = 146^{2} + \left(115\sqrt{2}\right)^{2}$

$y = 218.55$

$= 219$ m

(ii) Area of a triangular face

$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$

$A = 63.4$

$\text{Area} = \dfrac{1}{2}ab\,\sin C$

$= \dfrac{1}{2}(219)(219)\,\sin 63.4$

End of lesson

3D Diagrams — HL · Mathslive.ie