1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs

CAIE P2 2020 June Q6

8 marks Standard +0.3

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ where \(a\) is a constant. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
3. Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

CAIE P2 2024 June Q7

10 marks Standard +0.3

7

1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}

CAIE P2 2021 March Q2

5 marks Standard +0.3

2 Solve the equation \(\sec ^ { 2 } \theta \cot \theta = 8\) for \(0 < \theta < \pi\).

CAIE P2 2020 November Q1

3 marks Moderate -0.8

1 Solve the equation \(7 \cot \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

CAIE P2 2021 November Q6

10 marks Standard +0.3

6 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 8 x + 15 \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + b x + 18$$ where \(a\) and \(b\) are constants.

1. Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Given that the remainder is 40 when \(\mathrm { g } ( x )\) is divided by \(( x - 2 )\), find the value of \(b\).
3. When \(a\) and \(b\) have these values, factorise \(\mathrm { f } ( x ) - \mathrm { g } ( x )\) completely.
4. Hence solve the equation \(\mathrm { f } ( \operatorname { cosec } \theta ) - \mathrm { g } ( \operatorname { cosec } \theta ) = 0\) for \(0 < \theta < 2 \pi\).

CAIE P2 2022 November Q3

4 marks Standard +0.3

3 It is given that \(\sec \theta = \sqrt { 17 }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
Find the exact value of \(\tan \left( \theta + \frac { 1 } { 4 } \pi \right)\).

CAIE P2 2022 November Q1

4 marks Moderate -0.3

1 Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

CAIE P2 2023 November Q2

5 marks Standard +0.3

2 Solve the equation \(\sec \theta \cos \left( \theta - 60 ^ { \circ } \right) = 4\) for \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

CAIE P2 2024 November Q4

8 marks Standard +0.3

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
   
3. Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

CAIE P2 2005 June Q5

9 marks Moderate -0.3

5

1. By differentiating \(\frac { 1 } { \cos \theta }\), show that if \(y = \sec \theta\) then \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \sec \theta \tan \theta\).
2. The parametric equations of a curve are $$x = 1 + \tan \theta , \quad y = \sec \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin \theta\).
3. Find the coordinates of the point on the curve at which the gradient of the curve is \(\frac { 1 } { 2 }\).

CAIE P2 2007 June Q5

8 marks Standard +0.3

5

1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - x$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 1.0 and 1.2.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

CAIE P2 2009 June Q5

6 marks Standard +0.3

5 Solve the equation \(\sec x = 4 - 2 \tan ^ { 2 } x\), giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

CAIE P2 2010 June Q8

9 marks Standard +0.3

8

1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

CAIE P2 2013 June Q6

8 marks Standard +0.3

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 4 x - 2$$ where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.9\).
3. Show that this root also satisfies the equation $$x = \frac { 1 + 2 \tan x } { 4 \tan x }$$
4. Use the iterative formula \(x _ { n + 1 } = \frac { 1 + 2 \tan x _ { n } } { 4 \tan x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

CAIE P2 2014 June Q8

9 marks Standard +0.8

8 \includegraphics[max width=\textwidth, alt={}, center]{de8af872-9f77-4787-8e66-ed199405ca25-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

CAIE P2 2014 June Q8

9 marks Standard +0.8

8 \includegraphics[max width=\textwidth, alt={}, center]{22ba6cc7-7375-434e-9eaa-d536684dd727-3_581_650_1272_744} The diagram shows the curve $$y = \tan x \cos 2 x , \text { for } 0 \leqslant x < \frac { 1 } { 2 } \pi$$ and its maximum point \(M\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } x - \sec ^ { 2 } x - 2\).
2. Hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.

CAIE P2 2015 June Q6

10 marks Standard +0.3

6

1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
2. Hence
   1. solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
   2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

CAIE P2 2016 June Q4

8 marks Standard +0.3

4

1. Show that \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta\).
2. Hence
   1. find the exact value of \(\sin 105 ^ { \circ } + \sin 165 ^ { \circ }\),
   2. solve the equation \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

CAIE P2 2018 June Q7

11 marks Challenging +1.2

7

1. Express \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
2. Using your answer from part (i), solve the equation $$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$ for \(0 < \theta < 2 \pi\).
3. Find \(\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P2 2018 June Q7

10 marks Standard +0.8

7

1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE P3 2004 June Q1

3 marks Moderate -0.8

1 Sketch the graph of \(y = \sec x\), for \(0 \leqslant x \leqslant 2 \pi\).

CAIE P3 2005 June Q7

8 marks Standard +0.3

7

1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify, by calculation, that this root lies between 0.5 and 1 .
3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

CAIE P3 2011 June Q6

7 marks Standard +0.3

6

1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.5 and 0.8.
3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

CAIE P3 2012 June Q4

6 marks Standard +0.8

4 Solve the equation $$\operatorname { cosec } 2 \theta = \sec \theta + \cot \theta$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

CAIE P3 2014 June Q8

9 marks Standard +0.3

8

1. By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation $$\operatorname { cosec } x = x ( \pi - x )$$ has exactly two real roots in the interval \(0 < x < \pi\).
2. Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
3. The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
   1. Use the iterative formula $$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$ to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   2. Deduce the value of \(\beta\) correct to 2 decimal places.