1.05m Geometric proofs: of trig sum and double angle formulae

1. (a) Using the identity for \(\cos ( A + B )\), prove that

$$\cos 2 A \equiv 2 \cos ^ { 2 } A - 1$$ (b) Hence, using algebraic integration, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 8 } } \left( 5 - 4 \cos ^ { 2 } 3 x \right) d x$$

9 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).

8 The equation of a curve, in polar coordinates, is $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

1. \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-3_268_796_1567_717} The diagram shows the part of the curve for which \(0 \leqslant \theta \leqslant \alpha\), where \(\theta = \alpha\) is the equation of the tangent to the curve at \(O\). Find \(\alpha\) in terms of \(\pi\).
2. (a) If \(\mathrm { f } ( \theta ) = 1 - \sin 2 \theta\), show that \(\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )\) for all \(\theta\), where \(k\) is an integer.
   (b) Hence state the equations of the lines of symmetry of the curve $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
3. Sketch the curve with equation $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$ State the maximum value of \(r\) and the corresponding values of \(\theta\).

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

14 Some students are trying to prove an identity for \(\sin ( A + B )\). They start by drawing two right-angled triangles \(O D E\) and \(O E F\), as shown. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle \(D O E = A\) and angle \(E O F = B\).
In triangle OFR,
Line \(1 \quad \sin ( A + B ) = \frac { R F } { O F }\) Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 \(=\) \(\_\_\_\_\) \(+ \cos A \sin B\) 14

1. Explain why \(\frac { P F } { E F } \times \frac { E F } { O F }\) in Line 4 leads to \(\cos A \sin B\) in Line 5
   14
2. Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = \(+ \cos A \sin B\) 14
3. Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles. 14
4. Another student claims that by replacing \(B\) with \(- B\) in the identity for \(\sin ( A + B )\) it is possible to find an identity for \(\sin ( A - B )\). Assuming the identity for \(\sin ( A + B )\) is correct for all values of \(A\) and \(B\), prove a similar result for \(\sin ( A - B )\).

11

1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.