Exact trigonometric values

1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).

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 \(\sin A\).

4. The diagram shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

3

1. Use the formula for \(\sin ( \theta + \phi )\), with \(\theta = 45 ^ { \circ }\) and \(\phi = 60 ^ { \circ }\), to show that \(\sin 105 ^ { \circ } = \frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }\).
2. In triangle ABC , angle \(\mathrm { BAC } = 45 ^ { \circ }\), angle \(\mathrm { ACB } = 30 ^ { \circ }\) and \(\mathrm { AB } = 1\) unit (see Fig. 3). Fig. 3 Using the sine rule, together with the result in part (i), show that \(\mathrm { AC } = \frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }\).

7

1. 1. Given that \(\alpha\) is the acute angle such that \(\tan \alpha = \frac { 2 } { 5 }\), find the exact value of \(\cos \alpha\).
   2. Given that \(\beta\) is the obtuse angle such that \(\sin \beta = \frac { 3 } { 7 }\), find the exact value of \(\cos \beta\).
2. \includegraphics[max width=\textwidth, alt={}, center]{f25e2580-ba0b-42ce-bf86-63f2c2075223-3_316_662_955_700} The diagram shows a triangle \(A B C\) with \(A C = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A B C = \gamma\). Find the exact value of \(\sin \gamma\), simplifying your answer.

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.

5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,

1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
2. show that \(Q R = 2 \sqrt { 6 }\),
3. find \(\angle P Q R\) in degrees to 1 decimal place.

4 The triangle \(A B C\), shown in the diagram, is such that \(B C = 6 \mathrm {~cm} , A C = 5 \mathrm {~cm}\) and \(A B = 4 \mathrm {~cm}\). The angle \(B A C\) is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-3_442_652_452_678}

1. Use the cosine rule to show that \(\cos \theta = \frac { 1 } { 8 }\).
2. Hence use a trigonometrical identity to show that \(\sin \theta = \frac { 3 \sqrt { 7 } } { 8 }\).
3. Hence find the area of the triangle \(A B C\).

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 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]