1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

2

1. Show that \(\cos ( \alpha + \beta ) = \frac { 1 - \tan \alpha \tan \beta } { \sec \alpha \sec \beta }\).
2. Hence show that \(\cos 2 \alpha = \frac { 1 - \tan ^ { 2 } \alpha } { 1 + \tan ^ { 2 } \alpha }\).
3. Hence or otherwise solve the equation \(\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

3 Show that the equation \(\operatorname { cosec } x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$ Hence solve the equation for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to 1 decimal place.

2 Show that the equation \(\operatorname { cosec } x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$ Hence solve the equation for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to 1 decimal place.

5 Prove that \(\cot \beta - \cot \alpha = \frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta }\).

8

1. Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
2. Express \(\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }\) in partial fractions.
3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x\), expressing your answer in an exact form.

7.The variable \(y\) is defined by $$y = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \text { for } 0 < x < \frac { \pi } { 2 } .$$ A student was asked to prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - 4 \cot 2 x .$$ The attempted proof was as follows: $$\begin{aligned} y & = \ln \left( \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x \right) \\ & = \ln \left( \sec ^ { 2 } x \right) + \ln \left( \operatorname { cosec } ^ { 2 } x \right) \\ & = 2 \ln \sec x + 2 \ln \operatorname { cosec } x \\ \frac { \mathrm {~d} y } { \mathrm {~d} x } & = 2 \tan x - 2 \cot x \\ & = \frac { 2 \left( \sin ^ { 2 } x - \cos ^ { 2 } x \right) } { \sin x \cos x } \\ & = \frac { - 2 \cos 2 x } { \frac { 1 } { 2 } \sin 2 x } \\ & = - 4 \cot 2 x \end{aligned}$$

1. Identify the error in this attempt at a proof.
2. Give a correct version of the proof.
3. Find and simplify a general relationship between \(p\) and \(q\) ,where \(p\) and \(q\) are variables that depend on \(x\) ,such that the student would obtain the correct result when differentiating \(\ln ( p + q )\) with respect to \(x\) by the above incorrect method.
4. Given that \(p ( x ) = k \sec r x\) and \(q ( x ) = \operatorname { cosec } ^ { 2 } x\) ,where \(k\) and \(r\) are positive integers,find the values of \(k\) and \(r\) such that \(p\) and \(q\) satisfy the relationship found in part(c). \section*{END} Marks for presentation: 7
   TOTAL MARKS: 100

4.(a)Prove the identity $$( \sin x + \cos y ) \cos ( x - y ) \equiv ( 1 + \sin ( x - y ) ) ( \cos x + \sin y )$$ (b)Hence,or otherwise,show that $$\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } \equiv \frac { 1 + \tan \theta } { 1 - \tan \theta }$$ (c)Given that \(k > 1\) ,show that the equation \(\frac { \sin 5 \theta + \cos 3 \theta } { \cos 5 \theta + \sin 3 \theta } = k\) has a unique solution in the interval \(0 < \theta < \frac { \pi } { 4 }\)

3.(a)Use the formulae for \(\sin ( A \pm B )\) and \(\cos ( A \pm B )\) to prove that \(\tan \left( 90 ^ { \circ } - \theta \right) \equiv \cot \theta\) (b)Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 - \sec ^ { 2 } \left( \theta + 11 ^ { \circ } \right) = 2 \tan \left( \theta + 11 ^ { \circ } \right) \tan \left( \theta - 34 ^ { \circ } \right)$$ Give each answer as an integer in degrees.

2.Find the values of \(\tan \theta\) such that $$2 \sin ^ { 2 } \theta - \sin \theta \sec \theta = 2 \sin 2 \theta - 2 .$$

1.(a)Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ,in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) .
(b)Hence,or otherwise,show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid.
(4)
Find
(c) $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \\ & 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ (d)

3.The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ,satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$

1. Show that \(\tan \theta = 3 ^ { p }\) ,where \(p\) is a rational number to be found.
2. Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)

8. (a) Prove that $$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta < \frac { \pi } { 2 }\)

1. \(\sin 2 \theta - \tan \theta = \sqrt { 3 } \cos 2 \theta\)
2. \(\tan ( \theta + 1 ) \cos ( 2 \theta + 2 ) - \sin ( 2 \theta + 2 ) = 2\) Give your answers in radians to 3 significant figures, as appropriate.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

1

1. Show that the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ can be expressed in the form $$2 \cos ^ { 2 } x + 5 \cos x - 3 = 0$$
2. Hence solve the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(3 \tan 2 x = 1\)
2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)

9

1. Sketch the graph of \(y = \tan \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\) on the axes provided.
   On the same axes, sketch the graph of \(y = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\), indicating the point of intersection with the \(y\)-axis.
2. Show that the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) can be expressed in the form $$3 \sin ^ { 2 } \left( \frac { 1 } { 2 } x \right) + \sin \left( \frac { 1 } { 2 } x \right) - 3 = 0$$ Hence solve the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).

5

1. Show that the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\) can be expressed in the form $$6 \cos ^ { 2 } x - \cos x - 2 = 0 .$$
2. Hence solve the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

5 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(\sin 2 x = 0.5\)
2. \(2 \sin ^ { 2 } x = 2 - \sqrt { 3 } \cos x\)

7

1. Show that \(\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv \tan ^ { 2 } x - 1\).
2. Hence solve the equation $$\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5 - \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4 Solve the equation $$4 \cos ^ { 2 } x + 7 \sin x - 7 = 0$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

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.

9 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ).
2. Show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.
3. Solve the equation $$4 \cos ^ { 3 } \theta - 7 \cos \theta - 3 = 0$$ for \(0 \leqslant \theta \leqslant 2 \pi\). Give each solution for \(\theta\) in an exact form.

4

1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9 A curve has equation \(y = \sin ( a x )\), where \(a\) is a positive constant and \(x\) is in radians.

1. State the period of \(y = \sin ( a x )\), giving your answer in an exact form in terms of \(a\).
2. Given that \(x = \frac { 1 } { 5 } \pi\) and \(x = \frac { 2 } { 5 } \pi\) are the two smallest positive solutions of \(\sin ( a x ) = k\), where \(k\) is a positive constant, find the values of \(a\) and \(k\).
3. Given instead that \(\sin ( a x ) = \sqrt { 3 } \cos ( a x )\), find the two smallest positive solutions for \(x\), giving your answers in an exact form in terms of \(a\). \section*{END OF QUESTION PAPER}

8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$