1.05b Sine and cosine rules: including ambiguous case

6 In Fig. 6, OAB is a thin bent rod, with \(\mathrm { OA } = a\) metres, \(\mathrm { AB } = b\) metres and angle \(\mathrm { OAB } = 120 ^ { \circ }\). The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E . \begin{figure}[h] \end{figure}

1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a \sin \theta + b \sin \left( \theta - 60 ^ { \circ } \right) .$$
2. Hence show that \(h = \left( a + \frac { 1 } { 2 } b \right) \sin \theta - \frac { \sqrt { 3 } } { 2 } b \cos \theta\). The rod now rotates about O , so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.
3. Show that OB is horizontal when \(\tan \theta = \frac { \sqrt { 3 } b } { 2 a + b }\). In the case when \(a = 1\) and \(b = 2 , h = 2 \sin \theta - \sqrt { 3 } \cos \theta\).
4. Express \(2 \sin \theta - \sqrt { 3 } \cos \theta\) in the form \(R \sin ( \theta - \alpha )\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\).

6 In Fig. 6, \(\mathrm { ABC } , \mathrm { ACD }\) and AED are right-angled triangles and \(\mathrm { BC } = 1\) unit. Angles CAB and CAD are \(\theta\) and \(\phi\) respectively. \begin{figure}[h] \end{figure}

1. Find AC and AD in terms of \(\theta\) and \(\phi\).
2. Hence show that \(\mathrm { DE } = 1 + \frac { \tan \phi } { \tan \theta }\). Section B (36 marks)

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

1 In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).

1. Calculate the length \(B C\). \(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
2. Calculate the possible values of the angle \(A D B\).

6. Figure 1 Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = 2 x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C\) is \(18 \sqrt { 3 } \mathrm {~cm} ^ { 2 }\)

1. show that \(x = 2 \sqrt { 3 }\)
2. Hence find the exact length of BC, giving your answer as a simplified surd.

5. \begin{figure}[h] \end{figure} Figure 1 shows the design for a structure used to support a roof.
The structure consists of four steel beams, \(A B , B D , B C\) and \(A D\).
Given \(A B = 12 \mathrm {~m} , B C = B D = 7 \mathrm {~m}\) and angle \(B A C = 27 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\). The steel beams can only be bought in whole metre lengths.
2. Find the minimum length of steel that needs to be bought to make the complete structure.

4. Figure 1 Figure 1 shows a sketch of triangle \(A B C\) with \(A B = ( x + 2 ) \mathrm { cm } , B C = ( 3 x + 10 ) \mathrm { cm }\), \(A C = 7 x \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = \theta ^ { \circ }\)

1. 1. Show that \(17 x ^ { 2 } - 35 x - 48 = 0\)
   2. Hence find the value of \(x\).
2. Hence find the value of \(\theta\) giving your answer to one decimal place.

3. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing the position of three phone masts, \(A , B\) and \(C\).
The masts are identical and their bases are assumed to lie in the same horizontal plane.
From mast \(C\)

• mast \(A\) is 8.2 km away on a bearing of \(072 ^ { \circ }\)
• mast \(B\) is 15.6 km away on a bearing of \(039 ^ { \circ }\)
  1. Find the distance between masts \(A\) and \(B\), giving your answer in km to one decimal place.

An engineer needs to travel from mast \(A\) to mast \(B\).

Give a reason why the answer to part (a) is unlikely to be an accurate value for the distance the engineer travels.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of triangle \(A B D\) and triangle \(B C D\) Given that

• \(A D C\) is a straight line
• \(B D = ( x + 3 ) \mathrm { cm }\)
• \(B C = x \mathrm {~cm}\)
• angle \(B D C = 30 ^ { \circ }\)
• angle \(B C D = 140 ^ { \circ }\)
  1. show that \(x = 10.5\) correct to 3 significant figures.

Given also that \(A D = ( x - 2 ) \mathrm { cm }\)

find the length of \(A B\), giving your answer to 3 significant figures.

1. A parallelogram \(P Q R S\) has area \(50 \mathrm {~cm} ^ { 2 }\)

Given

• \(P Q\) has length 14 cm
• \(Q R\) has length 7 cm
• angle \(S P Q\) is obtuse
  find
  1. the size of angle \(S P Q\), in degrees, to 2 decimal places,
  2. the length of the diagonal \(S Q\), in cm , to one decimal place.

1 In the triangle \(A B C , A B = 3 , B C = 4\) and angle \(A B C = 30 ^ { \circ }\). Find the following.

1. The area of the triangle.
2. The length \(A C\).
3. The angle \(A C B\).

1 In triangle \(A B C , A B = 20 \mathrm {~cm}\) and angle \(B = 45 ^ { \circ }\).

1. Given that \(A C = 16 \mathrm {~cm}\), find the two possible values for angle \(C\), correct to 1 decimal place.
2. Given instead that the area of the triangle is \(75 \sqrt { 2 } \mathrm {~cm} ^ { 2 }\), find \(B C\).

1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 9.5 cm . The angle \(A O B\) is \(25 ^ { \circ }\).

1. Calculate the length of the straight line \(A B\).
2. Find the area of the segment shaded in the diagram.

4 A triangle ABC has sides \(\mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 9 \mathrm {~cm}\) and \(\mathrm { BC } = 10 \mathrm {~cm}\).

1. Find the cosine of angle BAC, giving your answer as a fraction in its lowest terms.
2. Find the exact area of the triangle.

1 The triangle ABC has an obtuse angle at A . The angle at B is \(15 ^ { \circ }\). The length of AC is 10 cm and the length of BC is 13 cm . Calculate the size of the angle at A .

4 The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the midpoint of AC . The angles BAC and BDC are each equal to \(\chi ^ { \circ }\). The lengths of AB and CD are 4 cm and 7 cm respectively. \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-3_606_529_1370_244} Determine the value of \(x\).

2 Fig. 2 shows a quadrilateral ABCD . The lengths AB and BC are 5 cm and 6 cm respectively. The angles \(\mathrm { ABC } , \mathrm { ACD }\) and DAC are \(60 ^ { \circ } , 60 ^ { \circ }\) and \(75 ^ { \circ }\) respectively. \begin{figure}[h] \end{figure} Calculate the exact value of the length AD.

5 The diagram shows the triangle ABC in which \(\mathrm { AC } = 13 \mathrm {~cm}\) and AB is the shortest side. The perimeter of the triangle is 32 cm . The area is \(24 \mathrm {~cm} ^ { 2 }\) and \(\sin \mathrm { B } = \frac { 4 } { 5 }\). Determine the lengths of AB and BC .

5 A triangular field has sides of length \(100 \mathrm {~m} , 120 \mathrm {~m}\) and 135 m .

1. Find the area of the field.
2. Explain why it would not be reasonable to expect your answer in (a) to be accurate to the nearest square metre.

5 A sphere of mass 3 kg hangs on a string. A horizontal force of magnitude \(F \mathrm {~N}\) acts on the sphere so that it hangs in equilibrium with the string making an angle of \(25 ^ { \circ }\) to the vertical. The force diagram for the sphere is shown below. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-05_502_513_408_244}

1. Sketch the triangle of forces for these forces.
2. Hence or otherwise determine each of the following:
   Answer all the questions.
   Section B (76 marks)

10 A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins \(B\) and \(C\). \(A B\) is a 30 cm rod and \(A C\) is a 15 cm rod. The angle \(B A C\) is \(\theta\) radians as shown in the diagram. The angle \(\theta\) increases at a rate of 0.1 radians per second.
Determine the rate of change of the length BC when \(\theta = \frac { 1 } { 3 } \pi\).

2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]