1.05b Sine and cosine rules: including ambiguous case

\includegraphics{figure_2} The diagram shows triangle \(ABC\) in which angle \(A\) is \(60°\) and the lengths of \(AB\) and \(AC\) are \((4 + h)\) cm and \((4 - h)\) cm respectively.

1. Show that the length of \(BC\) is \(p\) cm where $$p^2 = 16 + 3h^2.$$ [2]
2. Hence show that, when \(h\) is small, \(p \approx 4 + \lambda h^2 + \mu h^4\), where \(\lambda\) and \(\mu\) are rational numbers whose values are to be determined. [4]

A triangular field of grass, \(ABC\), has boundaries with lengths as follows: $$AB = 234 \text{ m} \qquad BC = 225 \text{ m} \qquad AC = 310 \text{ m}$$ The field is shown in the diagram below. \includegraphics{figure_7}

1. Find angle \(A\) [2 marks]
2. Farmers calculate the number of sheep they can keep in a field, by allowing one sheep for every \(1200 \text{ m}^2\) of grass. Find the maximum number of sheep which can be kept in the field \(ABC\) [3 marks]

\includegraphics{figure_1} A triangular lawn is modelled by the triangle \(ABC\), shown in Figure 1. The length \(AB\) is to be \(30\text{m}\) long. Given that angle \(BAC = 70°\) and angle \(ABC = 60°\),

1. calculate the area of the lawn to \(3\) significant figures. [4]
2. Why is your answer unlikely to be accurate to the nearest square metre? [1]

In a triangle \(PQR\), \(PQ = 20\) cm, \(PR = 10\) cm and angle \(QPR = \theta\), where \(\theta\) is measured in degrees. The area of triangle \(PQR\) is 80 cm\(^2\).

1. Show that the two possible values of \(\cos \theta = \pm \frac{3}{5}\) [4]

Given that \(QR\) is the longest side of the triangle,

1. find the exact perimeter of the triangle \(PQR\), giving your answer as a simplified surd. [3]

In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}

1. Determine the following, in either order.
   [9]

After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.

1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]

The diagram below shows a triangle \(ABC\) with \(AC = 5\) cm, \(AB = x\) cm, \(BC = y\) cm and angle \(BAC = 120°\). The area of the triangle \(ABC\) is \(14\) cm\(^2\). \includegraphics{figure_14} Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. [6]

The diagram below shows a triangle \(ABC\). \includegraphics{figure_6} Given that \(AB = 3\), \(BC = 2\sqrt{5}\), \(AC = 4 + \sqrt{3}\), find the value of \(\cos ABC\). Show all your working and give your answer in the form \(\frac{(a - b\sqrt{3})}{6\sqrt{5}}\), where \(a\), \(b\) are integers. [7]

In triangle \(ABC\), \(BC = 12\) cm and \(\cos ABC = \frac{2}{3}\). The length of \(AC\) is 2 cm greater than the length of \(AB\).

1. Find the lengths of \(AB\) and \(AC\). [4]
2. Find the exact value of \(\sin BAC\). Give your answer in its simplest form. [3]

In the triangle \(PQR\), \(PQ = 6\), \(PR = k\), \(P\hat{Q}R = 30°\).

1. For the case \(k = 4\), find the two possible values of \(QR\) exactly. [3]
2. Determine the value(s) of \(k\) for which the conditions above define a unique triangle. [3]

\includegraphics{figure_5} The diagram shows triangle \(ABC\), with \(AB = x\) cm, \(AC = (x + 2)\) cm, \(BC = 2\sqrt{7}\) cm and angle \(CAB = 60°\).

1. Find the value of \(x\). [4]
2. Find the area of triangle \(ABC\), giving your answer in an exact form as simply as possible. [2]

The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}

1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
2. Find the area of the shaded segment. [3]
3. Find the perimeter of the shaded segment. [4]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]

\includegraphics{figure_2} The shape \(AOCBA\), shown in Figure 2, consists of a sector \(AOB\) of a circle centre \(O\) joined to a triangle \(BOC\). The points \(A\), \(O\) and \(C\) lie on a straight line with \(AO = 7.5\) cm and \(OC = 8.5\) cm. The size of angle \(AOB\) is 1.2 radians. Find, in cm, the perimeter of the shape \(AOCBA\), giving your answer to one decimal place. [5]

In a triangle \(ABC\), \(AB = 9\) cm, \(BC = 7\) cm and \(AC = 4\) cm.

1. Show that \(\cos CAB = \frac{2}{3}\). [2]
2. Hence find the exact value of \(\sin CAB\). [2]
3. Find the exact area of triangle \(ABC\). [2]

In this question you must show detailed reasoning. The diagram shows triangle \(ABC\). \includegraphics{figure_8} The angles \(CAB\) and \(ABC\) are each \(45°\), and angle \(ACB = 90°\). The points \(D\) and \(E\) lie on \(AC\) and \(AB\) respectively. \(AE = DE = 1\), \(DB = 2\). Angle \(BED = 90°\), angle \(EBD = 30°\) and angle \(DBC = 15°\).

1. Show that \(BC = \frac{\sqrt{2} + \sqrt{6}}{2}\). [3]
2. By considering triangle \(BCD\), show that \(\sin 15° = \frac{\sqrt{6} - \sqrt{2}}{4}\). [3]