1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2

8 Functions f and g are defined for \(0 \leqslant x \leqslant 2 \pi\) by \(\mathrm { f } ( x ) = 2 \tan x\) and \(\mathrm { g } ( x ) = \sec x\).

1. 1. State the range of f .
   2. State the range of \(g\).
2. 1. Show that \(\operatorname { fg } ( 0.6 ) = 5.33\), correct to 3 significant figures.
   2. Explain why \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( 0.6 )\) is not defined.
3. In this question you must show detailed reasoning. Solve the equation \(( \mathrm { f } ( x ) ) ^ { 2 } + 6 \mathrm {~g} ( x ) = 0\).

4

1. Write \(2 x ^ { 2 } + 6 x + 7\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are constants.
2. State the coordinates of the minimum point on the graph of \(y = 2 x ^ { 2 } + 6 x + 7\).
3. Hence deduce

12. a. Prove that $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } \equiv \sin ^ { 2 } x$$ b. Hence solve, for \(- 360 ^ { \circ } < x < 360 ^ { \circ }\), the equation $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } = \frac { \cos 2 x } { 2 }$$

15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\operatorname { cosec } \theta - \sin \theta \equiv \cos \theta \cot \theta \quad \theta \neq ( 180 n ) ^ { \circ } \quad n \in \mathbb { Z }$$
2. Hence, or otherwise, solve for \(0 < x < 180 ^ { \circ }\) $$\operatorname { cosec } x - \sin x = \cos x \cot \left( 3 x - 50 ^ { \circ } \right)$$

6. $$f ( x ) = - 3 x ^ { 3 } + 8 x ^ { 2 } - 9 x + 10 , \quad x \in \mathbb { R }$$

1. 1. Calculate f(2)
   2. Write \(\mathrm { f } ( x )\) as a product of two algebraic factors. Using the answer to (a)(ii),
2. prove that there are exactly two real solutions to the equation $$- 3 y ^ { 6 } + 8 y ^ { 4 } - 9 y ^ { 2 } + 10 = 0$$
3. deduce the number of real solutions, for \(7 \pi \leqslant \theta < 10 \pi\), to the equation $$3 \tan ^ { 3 } \theta - 8 \tan ^ { 2 } \theta + 9 \tan \theta - 10 = 0$$

1. (a) Prove that

$$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.

1. (a) Prove

$$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } \equiv 2 \cot 2 \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } = 4$$ giving any solutions to one decimal place.

1. The curve \(C\) has parametric equations

$$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\)

1. A curve \(C\) is given by the equation

$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

3 In this question you must show detailed reasoning.
Solve the equation \(\sec ^ { 2 } \theta + 2 \tan \theta = 4\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

6

1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
   Section B (75 marks)

10 A particle \(B\), of mass 3 kg , moves in a straight line and has velocity \(v \mathrm {~ms} ^ { - 1 }\).
At time \(t\) seconds, where \(0 \leqslant t < \frac { 1 } { 4 } \pi\), a variable force of \(- ( 15 \sin 4 \mathrm { t } + 6 \mathrm { v } \tan 2 \mathrm { t } )\) Newtons is applied to \(B\). There are no other forces acting on \(B\). Initially, when \(t = 0 , B\) has velocity \(4.5 \mathrm {~ms} ^ { - 1 }\). The motion of \(B\) can be modelled by the differential equation \(\frac { d v } { d t } + P ( t ) v = Q ( t )\) where \(P ( t )\) and \(\mathrm { Q } ( \mathrm { t } )\) are functions of \(t\).

1. Find the functions \(\mathrm { P } ( \mathrm { t } )\) and \(\mathrm { Q } ( \mathrm { t } )\).
2. Using an integrating factor, determine the first time at which \(B\) is stationary according to the model.

9

1. 1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. 1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
   2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}

7

1. Given that \(\frac { \cos ^ { 2 } x + 4 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 7\), show that \(\tan ^ { 2 } x = \frac { 3 } { 2 }\).
2. Hence solve the equation \(\frac { \cos ^ { 2 } 2 \theta + 4 \sin ^ { 2 } 2 \theta } { 1 - \sin ^ { 2 } 2 \theta } = 7\) in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), giving your values of \(\theta\) to the nearest degree.
   [0pt] [4 marks]

5

1. 1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
   2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
   (3 marks)

2

1. Solve the equation \(\cot x = 2\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
2. Show that the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\) can be written as $$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
3. Solve the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.

4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi \\ \mathrm {~g} ( x ) = | x | , & \text { for all real values of } x \end{array}$$

1. Find the range of f .
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\), giving your answer in an exact form.
3. 1. Write down an expression for \(\mathrm { gf } ( x )\).
   2. Sketch the graph of \(y = \operatorname { gf } ( x )\) for \(0 \leqslant x \leqslant 2 \pi\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos \frac { 1 } { 2 } x\).

7

1. Solve the equation \(\sec x = - 5\), giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
2. Show that the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ can be written in the form $$\sec ^ { 2 } x = 25$$
3. Hence, or otherwise, solve the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
   (3 marks)

4

1. By using a suitable trigonometrical identity, solve the equation $$\tan ^ { 2 } \theta = 3 ( 3 - \sec \theta )$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
2. Hence solve the equation $$\tan ^ { 2 } \left( 4 x - 10 ^ { \circ } \right) = 3 \left[ 3 - \sec \left( 4 x - 10 ^ { \circ } \right) \right]$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 90 ^ { \circ }\).

7 A curve has equation \(y = 4 x \cos 2 x\).

1. Find an exact equation of the tangent to the curve at the point on the curve where $$x = \frac { \pi } { 4 }$$
2. The region shaded on the diagram below is bounded by the curve \(y = 4 x \cos 2 x\) and the \(x\)-axis from \(x = 0\) to \(x = \frac { \pi } { 4 }\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_487_878_740_591} By using integration by parts, find the exact value of the area of the shaded region.
   (5 marks)
   \includegraphics[max width=\textwidth, alt={}]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_1275_1717_1432_150}

3 A curve is defined for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\) by the equation \(y = x \cos 2 x\), and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-3_757_878_402_559}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(A\), where \(x = \alpha\), on the curve is a stationary point.
   1. Show that \(1 - 2 \alpha \tan 2 \alpha = 0\).
   2. Show that \(0.4 < \alpha < 0.5\).
   3. Show that the equation \(1 - 2 x \tan 2 x = 0\) can be rearranged to become \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\).
   4. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find \(x _ { 3 }\), giving your answer to two significant figures.
3. Use integration by parts to find \(\int _ { 0 } ^ { 0.5 } x \cos 2 x \mathrm {~d} x\), giving your answer to three significant figures.

3

1. Solve the equation \(\tan x = - \frac { 1 } { 3 }\), giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
2. Show that the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ can be written in the form \(3 \tan ^ { 2 } x - 5 \tan x - 2 = 0\).
3. Hence, or otherwise, solve the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ giving all the values of \(x\) in the interval \(0 < x < 2 \pi\) in radians to two decimal places.
   (4 marks)

5

1. Show that the equation $$10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x$$ can be written in the form $$10 \cot ^ { 2 } x + 11 \cot x - 6 = 0$$
2. Hence, given that \(10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x\), find the possible values of \(\tan x\).