Find exact trigonometric values

1. (i) Show that

$$\sin ( x + 30 ) ^ { \circ } + \sin ( x - 30 ) ^ { \circ } \equiv a \sin x ^ { \circ }$$ where \(a\) is a constant to be found.
(ii) Hence find the exact value of \(\sin 75 ^ { \circ } + \sin 15 ^ { \circ }\), giving your answer in the form \(b \sqrt { 6 }\).

2

1. Prove the identity $$\sin \left( x + 30 ^ { \circ } \right) + ( \sqrt { } 3 ) \cos \left( x + 30 ^ { \circ } \right) \equiv 2 \cos x$$ where \(x\) is measured in degrees.
2. Hence express \(\cos 15 ^ { \circ }\) in surd form.

3 Using appropriate right-angled triangles, show that \(\tan 45 ^ { \circ } = 1\) and \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Hence show that \(\tan 75 ^ { \circ } = 2 + \sqrt { 3 }\).

1．（a）Write down the exact value of \(\cos 405 ^ { \circ }\)
（b）Hence，using a double angle identity for cosine，or otherwise，determine the exact value of \(\cos 101.25 ^ { \circ }\) ，giving your answer in the form $$a \sqrt { b + c \sqrt { 2 + \sqrt { 2 } } }$$ where \(a\) ，\(b\) and \(c\) are rational numbers．

2．Find the value of $$\arccos \left( \frac { 1 } { \sqrt { 2 } } \right) + \arcsin \left( \frac { 1 } { 3 } \right) + 2 \arctan \left( \frac { 1 } { \sqrt { 2 } } \right)$$ Give your answer as a multiple of \(\pi\) ． $$\text { (arccos } x \text { is an alternative notion for } \cos ^ { - 1 } x \text { etc.) }$$

8 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).

1. Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre \(\mathrm { O } . \mathrm { AB }\) is one of the sides of the polygon. \(C\) is the midpoint of \(A B\). Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_455_428_523_900} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{figure} (A) Show that \(\mathrm { AB } = 2 \sin 15 ^ { \circ }\).
   (B) Use a double angle formula to express \(\cos 30 ^ { \circ }\) in terms of \(\sin 15 ^ { \circ }\). Using the exact value of \(\cos 30 ^ { \circ }\), show that \(\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }\).
   (C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6 \sqrt { 2 - \sqrt { 3 } }\).
2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_456_428_1621_900} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} (A) Show that \(\mathrm { DE } = 2 \tan 15 ^ { \circ }\).
   (B) Let \(t = \tan 15 ^ { \circ }\). Use a double angle formula to express \(\tan 30 ^ { \circ }\) in terms of \(t\). Hence show that \(t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0\).
   (C) Solve this equation, and hence show that \(\pi < 12 ( 2 - \sqrt { 3 } )\).
3. Use the results in parts (i)( \(C\) ) and (ii)( \(C\) ) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form. \section*{ADVANCED GCE
   MATHEMATICS (MEI)} 4754B
   Applications of Advanced Mathematics (C4) Paper B: Comprehension Candidates answer on the question paper
   Monday 1 June 2009
   OCR Supplied Materials:
   Morning
   • Insert (inserted)
   • MEI Examination Formulae and Tables (MF2)
   Duration: Up to 1 hour
   Other Materials Required:
   • Rough paper
     \includegraphics[max width=\textwidth, alt={}, center]{b4861178-720d-4803-a608-abef350efb0e-5_122_442_1023_1370}
   1 On lines 90 and 91, the article says "The average score for each player works out to be 0.25 points per round". Derive this figure. 2 Line 47 gives the inequality \(b > c > d > w\).
   Interpret each of the following inequalities in the context of the example from the 1st World War.
4. \(b > w\)
5. \(c > d\)
6. \(\_\_\_\_\)
7. \(\_\_\_\_\)
   3 Table 3 illustrates a possible game where you always co-operate. In lines 98 and 99 the article says "Clearly the longer the game goes on the closer your average score approaches - 2 points per round and that of your opponent approaches 3 ." How many rounds have you played when your average score is - 1.999 ?
   4 A Prisoner's Dilemma game is proposed in which $$b = 6 , c = 1 , d = - 1 \text { and } w = - 3 .$$ Using the information in the article, state whether these values would allow long-term co-operation to evolve. Justify your answer.
   5 In a Prisoner's Dilemma game both players keep strictly to a Tit-for-tat strategy. You start with C and your opponent starts with D . The scoring system of \(b = 3 , c = 1 , d = - 1\) and \(w = - 2\) is used.
8. This table shows the first 8 out of many rounds. Complete the table.

   Round	You	Opponent	Your score	Opponent's score
   1	C	D
   2
   3
   4
   5
   6
   7
   8
   ⋯	⋯	⋯	⋯	⋯

9. Find your average score per round in the long run.
   6 In the article, the scoring system is \(b = 3 , c = 1 , d = - 1\) and \(w = - 2\). In Axelrod's experiment, negative numbers were avoided by taking \(b = 5 , c = 3 , d = 1\) and \(w = 0\). State the effect this change would have on
10. the players' scores,
11. who wins.
12. \(\_\_\_\_\)
13. \(\_\_\_\_\)
    7 Two companies, X and Y , are the only sellers of ice cream on an island. They both have a market share of about \(50 \%\). Although their ice cream is much the same, both companies spend a lot of money on advertising.
14. What agreement might the companies reach if they decide to co-operate?
15. What advantage would a company hope to gain by 'defecting' from this agreement?
    RECOGNISING ACHIEVEMENT

3 Using appropriate right-angled triangles, show that \(\tan 45 ^ { \circ } = 1\) and \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\). Hence show that \(\tan 75 ^ { \circ } = 2 + \sqrt { 3 }\).

14

1. Show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) \Rightarrow \arctan \left( \frac { 1 } { 2 } \right) + \arctan \left( \frac { 1 } { 3 } \right) = \arctan 1 .$$
2. Use the arctan addition formula in line 23 to show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) , \text { as given in line } 39 .$$

15 Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41 . \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

15 Fig. 15 shows a unit circle and the escribed regular polygon with 12 edges. \begin{figure}[h] \end{figure}

1. Show that the perimeter of the polygon is \(24 \tan 15 ^ { \circ }\).
2. Using the formula for \(\tan ( \theta - \phi )\) show that the perimeter of the polygon is \(48 - 24 \sqrt { 3 }\).

6 In this question you must show detailed reasoning.

1. Use the formula for \(\tan ( A - B )\) to show that \(\tan \frac { \pi } { 12 } = 2 - \sqrt { 3 }\).
2. Solve the equation \(2 \sqrt { 3 } \sin 3 A - 2 \cos 3 A = 1\) for \(0 ^ { \circ } \leqslant A < 180 ^ { \circ }\).

3 It is given that \(3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).

1. Find the value of \(R\).
2. Show that \(\alpha \approx 33.7 ^ { \circ }\).
3. Hence write down the maximum value of \(3 \cos \theta - 2 \sin \theta\) and find a positive value of \(\theta\) at which this maximum value occurs.