1.05b Sine and cosine rules: including ambiguous case

3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.

1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.

10 Ship \(A\) is 15 km due south of ship \(B\). Ship \(B\) is travelling at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) is travelling at \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find

1. the bearing, to the nearest degree, that \(A\) must take in order to get as close as possible to \(B\), [4]
2. the time, in minutes, that it takes for the ships to be as close as possible.

4 The diagram shows a triangle \(A B C\) in which \(A B = 5 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(B C A = 20 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4e774e5-76fd-48ff-9bce-a995b3ba517b-2_355_767_1695_689}

1. Find angle \(B A C\), given that it is obtuse.
2. Find the shortest distance from \(A\) to \(B C\).

1 The diagram shows the triangle \(A B C\). \(A B = 10 \mathrm {~cm} , A C = 7 \mathrm {~cm}\) and angle \(B A C = 100 ^ { \circ }\).

1. Find the length \(B C\).
2. Find the area of the triangle \(A B C\).

3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle \(A B C\) in which angle \(B = 39 ^ { \circ }\), angle \(C = 28 ^ { \circ } , A B = x \mathrm {~cm}\) and \(A C = ( 2 x - 1 ) \mathrm { cm }\). Find the value of \(x\).

3 A triangle \(A B C\) has sides \(A B , B C\) and \(C A\) of lengths \(7 \mathrm {~cm} , 6 \mathrm {~cm}\) and 8 cm respectively.

1. Show that \(\cos A B C = \frac { 1 } { 4 }\).
2. Find the area of triangle \(A B C\).

4 The diagram shows triangle \(A B C\), in which \(A B = 1\) unit , \(A C = k\) units and \(B C = 2\) units .

1. Express \(\cos C\) in terms of \(k\).
2. Given that \(\cos C < \frac { 7 } { 8 }\), show that \(2 k ^ { 2 } - 7 k + 6 < 0\) and find the set of possible values of \(k\).

The triangle \(A B C\) is such that \(A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}\) and \(A \widehat { B C } = 32 ^ { \circ }\). Find two possible values for the area of the triangle \(A B C\). a) Use the binomial theorem to expand \(( a + \sqrt { b } ) ^ { 4 }\).
b) Hence, deduce an expression in terms of \(a\) and \(b\) for \(( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }\). a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }\) and \(\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }\). Determine the range of values for \(\mu\) such that \(\mu | \mathbf { v } | > | \mathbf { u } |\).
b) The point \(A\) has position vector \(11 \mathbf { i } - 4 \mathbf { j }\) and the point \(B\) has position vector \(21 \mathbf { i } + \mathbf { j }\). Determine the position vector of the point \(C\), which lies between \(A\) and \(B\), such that \(A C : C B\) is \(2 : 3\). Find the values of \(m\) for which the equation \(4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )\) has real roots. [5]

\includegraphics{figure_6} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius \(8\) cm. The line \(AB\) is a chord of the circle and angle \(AOB = 2.4\) radians. Find

1. the length of \(AB\), [2]
2. the perimeter of the plate, [3]
3. the area of the plate. [3]

\includegraphics{figure_3} The diagram shows triangle \(ABC\) which is right-angled at \(A\). Angle \(ABC = \frac{1}{4}\pi\) radians and \(AC = 8\) cm. The points \(D\) and \(E\) lie on \(BC\) and \(BA\) respectively. The sector \(ADE\) is part of a circle with centre \(A\) and is such that \(BDC\) is the tangent to the arc \(DE\) at \(D\).

1. Find the length of \(AD\). [3]
2. Find the area of the shaded region. [3]

\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).

1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
2. Hence, showing all necessary working, calculate \(\theta\). [3]

\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.

1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
2. Find the perimeter of the shaded region. [4]

\includegraphics{figure_4} The triangle \(XYZ\) in Figure 4 has \(XY = 6\) cm, \(YZ = 9\) cm, \(ZX = 4\) cm and angle \(ZXY = a\). The point \(W\) lies on the line \(XY\). The circular arc \(ZW\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm.

1. Show that, to 3 significant figures, \(a = 2.22\) radians. [2]
2. Find the area, in cm\(^2\), of the major sector \(XZWX\). [3]

The region, shown shaded in Figure 4, is to be used as a design for a logo. Calculate

1. the area of the logo [3]
2. the perimeter of the logo. [4]

In the triangle \(ABC\), \(AB = 8\) cm, \(AC = 7\) cm, \(\angle ABC = 0.5\) radians and \(\angle ACB = x\) radians.

1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. [3]

Given that there are two possible values of \(x\),

1. find these values of \(x\), giving your answers to 2 decimal places. [3]

\includegraphics{figure_2} In Figure 2 \(OAB\) is a sector of a circle, radius 5 m. The chord \(AB\) is 6 m long.

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

\includegraphics{figure_1} Figure 1 shows the triangle \(ABC\), with \(AB = 6\) cm, \(BC = 4\) cm and \(CA = 5\) cm.

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

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

\includegraphics{figure_1} Figure 1 shows 3 yachts \(A\), \(B\) and \(C\) which are assumed to be in the same horizontal plane. Yacht \(B\) is 500 m due north of yacht \(A\) and yacht \(C\) is 700 m from \(A\). The bearing of \(C\) from \(A\) is 015°.

1. Calculate the distance between yacht \(B\) and yacht \(C\), in metres to 3 significant figures. [3]

The bearing of yacht \(C\) from yacht \(B\) is \(\theta°\), as shown in Figure 1.

1. Calculate the value of \(\theta\). [4]

\includegraphics{figure_1} Figure 1 shows a logo \(ABD\). The logo is formed from triangle \(ABC\). The mid-point of \(AC\) is \(D\) and \(BC = AD = DC = 6\) cm. \(\angle BCA = 0.4\) radians. The curve \(BD\) is an arc of a circle with centre \(C\) and radius 6 cm.

1. Write down the length of the arc \(BD\). [1]
2. Find the length of \(AB\). [3]
3. Write down the perimeter of the logo \(ABD\), giving your answer to 3 significant figures. [1]

\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]

Calculate,

1. the area, in cm\(^2\), of the sector \(APQ\), [2]
2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
3. the perimeter, in cm, of the shaded region \(BPQC\). [4]

The triangle \(ABC\), shown in the diagram, is such that \(AB = 7\) cm, \(AC = 5\) cm, \(BC = 8\) cm and angle \(ABC = \theta\). \includegraphics{figure_1}

1. Show that \(\theta = 38.2°\), correct to the nearest \(0.1°\). [3]
2. Calculate the area of triangle \(ABC\), giving your answer, in cm\(^2\), to three significant figures. [2]

\includegraphics{figure_1} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.

1. Find, in surd form, the length \(AB\). [2]
2. Find, in terms of \(\pi\), the area of the badge. [2]
3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [2]

\includegraphics{figure_3} Fig. 3 Triangle ABC has AB = 9 cm, BC = 10 cm and CA = 5 cm. A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, ∠BAC = 1.504 radians. [3]

Calculate,

1. the area, in cm², of the sector APQ, [2]
2. the area, in cm², of the shaded region BPQC, [3]
3. the perimeter, in cm, of the shaded region BPQC. [4]

END