Ambiguous case (two solutions)

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.

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}

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 \(A D B\).

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 \(B C\).

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]

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]

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]