Chapter 8

Two aircraft leave an airfield at the same time. One travels due north at an average speed of \(300 \mathrm{~km} / \mathrm{h}\) and the other due west at an average speed of \(220 \mathrm{~km} / \mathrm{h}\). Calculate their distance apart after fourhours.

If \(\cos X=\frac{9}{41}\) determine the value of the other five trigonometry ratios.

Point \(A\) lies at co-ordinate \((2,3)\) and point \(B\) at \((8,7)\). Determine (a) the distance \(A B\), (b) the gradient of the straight line \(A B\), and (c) the angle \(A B\) makes with the horizontal.

Evaluate, correct to 4 decimal places: (a) sine \(168^{\circ} 14^{\prime}\) (b) cosine \(271.41^{\circ}\) (c) tangent \(98^{\circ} 4^{\prime}\)

Evaluate, correct to 4 decimal places: (a) secant \(161^{\circ}\) (b) secant \(302^{\circ} 29^{\prime}\)

Evaluate, correct to 4 decimal places: (a) cotangent \(17.49^{\circ}\) (b) cotangent \(163^{\circ} 52^{\prime}\)

Evaluate, correct to 4 significant figures: (a) \(\sin 1.481\) (b) \(\cos (3 \pi / 5)\) (c) \(\tan 2.93\)

Evaluate, correct to 4 decimal places: (a) secant \(5.37\) (b) \(\operatorname{cosecant} \pi / 4\) (c) cotangent \(\pi / 24\)

Find, in degrees, the acute angle \(\sin ^{-1} 0.4128\) correct to 2 decimal places.

Find the acute angle \(\cos ^{-1} 0.2437\) in degrees and minutes.

Find the acute angle \(\tan ^{-1} 7.4523\) in degrees and minutes.

Determine the acute angles: (a) \(\sec ^{-1} 2.3164\) (b) \(\operatorname{cosec}^{-1} 1.1784\) (c) \(\cot ^{-1} 2.1273\)

A locomotive moves around a curve of radius, \(r=500 \mathrm{~m}\). The angle of banking, \(\theta\), is given by: \(\theta=\tan ^{-1}\left(\frac{v^{2}}{r g}\right)\) where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) and \(v\) is the speed in \(\mathrm{m} / \mathrm{s}\). Calculate the angle of banking when the speed of the locomotive is \(30 \mathrm{~km} / \mathrm{h}\).

Evaluate correct to 4 decimal places: (a) \(\sec \left(-115^{\circ}\right)\) (b) \(\operatorname{cosec}\left(-95^{\circ} 47^{\prime}\right)\)

Solve triangle \(\mathrm{XYZ}\) given \(\angle X=90^{\circ}, \angle Y=23^{\circ} 17^{\prime}\) and \(Y Z=20.0 \mathrm{~mm}\). Determine also its area.

An electricity pylon stands on horizontal ground. At a point \(80 \mathrm{~m}\) from the base of the pylon, the angle of elevation of the top of the pylon is \(23^{\square}\). Calculate the height of the pylon to the nearest metre.

A surveyor measures the angle of elevation of the top of a perpendicular building as \(19^{\square}\). He moves \(120 \mathrm{~m}\) nearer the building and finds the angle of elevation is now \(47 \square\). Determine the height of the building.

The angle of depression of a ship viewed at a particular instant from the top of a \(75 \mathrm{~m}\) vertical cliff is \(30 \square\). Find the distance of the ship from the base of the cliff at this instant. The ship is sailing away from the cliff at constant speed and one minute later its angle of depression from the top of the cliff is \(20^{\square}\). Determine the speed of the ship in \(\mathrm{km} / \mathrm{h}\).

In a triangle \(\mathrm{XYZ}, \angle X=51^{\circ}, \angle Y=67\) and \(Y Z=15.2 \mathrm{~cm}\). Solve the triangle and find its area.

Solve the triangle \(\mathrm{PQR}\) and find its area given that \(Q R=36.5 \mathrm{~mm}, P R=\) \(29.6 \mathrm{~mm}\) and \(\angle Q=36\).

Solve triangle DEF and find its area given that \(E F=35.0 \mathrm{~mm}\), \(D E=25.0 \mathrm{~mm}\) and \(\angle E=64^{\circ}\)

A triangle \(\mathrm{ABC}\) has sides \(a=9.0 \mathrm{~cm}, b=7.5 \mathrm{~cm}\) and \(c=6.5 \mathrm{~cm}\). Determine its three angles and its area.

A room \(8.0 \mathrm{~m}\) wide has a span roof which slopes at \(33 \square\) on one side and \(40^{\square}\) on the other. Find the length of the roof slopes, correct to the nearest centimetre.