Exercises 19 (Sine and cosine rules)

  1. For the simple crank mechanism \(ABC\), given that \(AB = 60\,\mathrm{mm}\) and \(BC = 170\,\mathrm{mm}\), calculate the distance \(AC\) when the angle at \(B\) is \(140^\circ\).

  2. A scalene triangle \(ABC\) has the lengths of the sides set out as \(AB = 65\,\mathrm{mm}\), \(AC = 95\,\mathrm{mm}\) and \(BC = 80\,\mathrm{mm}\). Determine to the nearest degree the value of each angle.

  3. Three holes are marked on a pitch circle with chordal distances as shown below. Determine the values for the angles \(BAC\) and \(ABC\) and the pitch circle diameter.

  4. A mass \(m\) is suspended by two wires from a horizontal beam. If \(PK=3.3\,\mathrm{m}\), \(KQ=4.7\,\mathrm{m}\) and \(PQ=5\,\mathrm{m}\), determine the angle \(PQK\) and the vertical distance of \(K\) below the beam.

  5. Calculate the coordinates \(x\) and \(y\) of the centre of the hole relative to the two axes shown.

Solutions: 1. \(219.4\,\mathrm{mm}\); 2. \(56.3^\circ\), \(42.6^\circ\), \(81.1^\circ\); 3. \(43.16^\circ\), \(30.1^\circ\), \(43.87\,\mathrm{mm}\); 4. \(39.6^\circ\), \(3.00\,\mathrm{m}\); 5. \((126,81.4)\)