1.05b Sine and cosine rules: including ambiguous case

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)

5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that

• \(B D = x \mathrm {~m}\)
• \(C D = ( 1 + x ) \mathrm { m }\)
• \(B C = 5 \mathrm {~m}\)
• angle \(B C D = \theta ^ { \circ }\)
  1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)

Given also that

12. Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find

1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
2. the perimeter of the platform to 3 significant figures,
3. the area of the platform to 3 significant figures.

9. \begin{figure}[h] \end{figure} In Figure 3, the points \(A\) and \(B\) are the centres of the circles \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The circle \(C _ { 1 }\) has radius 10 cm and the circle \(C _ { 2 }\) has radius 5 cm . The circles intersect at the points \(X\) and \(Y\), as shown in the figure. Given that the distance between the centres of the circles is 12 cm ,

1. calculate the size of the acute angle \(X A B\), giving your answer in radians to 3 significant figures,
2. find the area of the major sector of circle \(C _ { 1 }\), shown shaded in Figure 3,
3. find the area of the kite \(A Y B X\).

11. \begin{figure}[h] \end{figure} Figure 1 shows a triangle \(X Y Z\) with \(X Y = 10 \mathrm {~cm} , Y Z = 16 \mathrm {~cm}\) and \(Z X = 12 \mathrm {~cm}\).

1. Find the size of the angle \(Y X Z\), giving your answer in radians to 3 significant figures. The point \(A\) lies on the line \(X Y\) and the point \(B\) lies on the line \(X Z\) and \(A X = B X = 5 \mathrm {~cm} . A B\) is the arc of a circle with centre \(X\). The shaded region \(S\), shown in Figure 1, is bounded by the lines \(B Z , Z Y , Y A\) and the arc \(A B\). Find
2. the perimeter of the shaded region to 3 significant figures,
3. the area of the shaded region to 3 significant figures.

13. \begin{figure}[h] \end{figure} Figure 4 shows the position of two stationary boats, \(A\) and \(B\), and a port \(P\) which are assumed to be in the same horizontal plane. Boat \(A\) is 8.7 km on a bearing of \(314 ^ { \circ }\) from port \(P\).
Boat \(B\) is 3.5 km on a bearing of \(052 ^ { \circ }\) from port \(P\).

1. Show that angle \(A P B\) is \(98 ^ { \circ }\)
2. Find the distance of boat \(B\) from boat \(A\), giving your answer to one decimal place.
3. Find the bearing of boat \(B\) from boat \(A\), giving your answer to the nearest degree.

10. \begin{figure}[h] \end{figure} Figure 1 shows the design for a shop sign \(A B C D A\). The sign consists of a triangle \(A O D\) joined to a sector of a circle \(D O B C D\) with radius 1.8 m and centre \(O\). The points \(A , B\) and \(O\) lie on a straight line.
Given that \(A D = 3.9 \mathrm {~m}\) and angle \(B O D\) is 0.84 radians,

1. calculate the size of angle \(D A O\), giving your answer in radians to 3 decimal places.
2. Show that, to one decimal place, the length of \(A O\) is 4.9 m .
3. Find, in \(\mathrm { m } ^ { 2 }\), the area of the shop sign, giving your answer to one decimal place.
4. Find, in m , the perimeter of the shop sign, giving your answer to one decimal place.

1. \begin{figure}[h] \end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).

1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.

15. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle shown in Figure 4 has centre \(P ( 5,6 )\) and passes through the point \(A ( 12,7 )\). Find

1. the exact radius of the circle,
2. an equation of the circle,
3. an equation of the tangent to the circle at the point \(A\). The circle also passes through the points \(B ( 0,1 )\) and \(C ( 4,13 )\).
4. Use the cosine rule on triangle \(A B C\) to find the size of the angle \(B C A\), giving your answer in degrees to 3 significant figures.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a design for a triangular garden \(A B C\). The garden has sides \(B A\) with length \(10 \mathrm {~m} , B C\) with length 6 m and \(C A\) with length 12 m . The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 10 m . A flowerbed \(B C D\) is shown shaded in Figure 2.

1. Find the size of angle \(B A C\), in radians, to 4 decimal places.
2. Find the perimeter of the flowerbed \(B C D\), in m , to 2 decimal places.
3. Find the area of the flowerbed \(B C D\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

10. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 shows a semicircle with centre \(O\) and radius \(3 \mathrm {~cm} . X Y\) is the diameter of this semicircle. The point Z is on the circumference such that angle \(X O Z = 1.3\) radians. The shaded region enclosed by the chord \(X Z\), the arc \(Z Y\) and the diameter \(X Y\) is a template for a badge. Find, giving each answer to 3 significant figures,

1. the length of the chord \(X Z\),
2. the perimeter of the template \(X Z Y X\),
3. the area of the template.

8. \begin{figure}[h] \end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.

1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
2. Find the area of the compound shape, giving your answer to 3 significant figures.

15. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4 is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 4. Calculate
3. the area of this shaded region,
4. the perimeter \(Z W Y Z\) of this shaded region. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-39_90_54_2576_1868}

7. \begin{figure}[h] \end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), is an arc of a circle with centre \(A\) and radius 8 cm . The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the perimeter of \(R\), giving your answer to 3 significant figures,
3. the area of \(R\), giving your answer to 3 significant figures.

5. \begin{figure}[h] \end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.

1. Show that \(\cos A \hat { O } B = \frac { 7 } { 25 }\).
2. Hence find the angle \(A \hat { O } B\) in radians, giving your answer to 3 decimal places.
3. Calculate the area of the sector \(O A B\).
4. Hence calculate the shaded area.

4. \begin{figure}[h] \end{figure} An emblem, as shown in Figure 1, consists of a triangle \(A B C\) joined to a sector \(C B D\) of a circle with radius 4 cm and centre \(B\). The points \(A , B\) and \(D\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B D = 4 \mathrm {~cm}\). Angle \(B A C = 0.6\) radians and \(A C\) is the longest side of the triangle \(A B C\).

1. Show that angle \(A B C = 1.76\) radians, correct to 3 significant figures.
2. Find the area of the emblem.

2. In the triangle \(A B C , A B = 11 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and \(C A = 8 \mathrm {~cm}\).

1. Find the size of angle \(C\), giving your answer in radians to 3 significant figures.
2. Find the area of triangle \(A B C\), giving your answer in \(\mathrm { cm } ^ { 2 }\) to 3 significant figures.

7. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 1 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 1 is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 1. Calculate
3. the area of this shaded region,
4. the perimeter \(Z W Y Z\) of this shaded region.

7. In the triangle \(A B C , A B = 8 \mathrm {~cm} , A C = 7 \mathrm {~cm} , \angle A B C = 0.5\) radians and \(\angle A C B = x\) radians.

1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. Given that there are two possible values of \(x\),
2. find these values of \(x\), giving your answers to 2 decimal places.

4. Figure 1 Figure 1 shows the triangle \(A B C\), with \(A B = 6 \mathrm {~cm} , B C = 4 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\).

1. Show that \(\cos A = \frac { 3 } { 4 }\).
2. Hence, or otherwise, find the exact value of \(\sin A\).

8. \begin{figure}[h] \end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,

1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
   The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
2. find the amount of grass seed needed, giving your answer to the nearest 10 g .

5. \begin{figure}[h] \end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).

1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.

2. In the triangle \(A B C , A B = 16 \mathrm {~cm} , A C = 13 \mathrm {~cm}\), angle \(A B C = 50 ^ { \circ }\) and angle \(B C A = x ^ { \circ }\) Find the two possible values for \(x\), giving your answers to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{752efc6c-8d0e-46a6-b75d-5125956969d8-05_104_107_2631_1774}

9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).

1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
3. Find the maximum area of \(\triangle P Q R\).
4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END