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

3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).

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 \(( \mathrm { f } ( x ) ) ^ { 2 } + 6 \mathrm {~g} ( x ) = 0\).

1. (i) Solve \(0 \leq \theta \leq 180 ^ { 0 }\), the equation

$$4 \cos \theta = \sqrt { 3 } \operatorname { cosec } \theta$$ (ii) Solve, for \(0 \leq x \leq 2 \pi\), the equation $$\cos x - \sqrt { 3 } \sin x = \sqrt { 3 }$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\frac { 1 } { \operatorname { cosec } \theta - 1 } + \frac { 1 } { \operatorname { cosec } \theta + 1 } \equiv 2 \tan \theta \sec \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$
2. Hence solve, for \(0 < x < 90 ^ { \circ }\), the equation $$\frac { 1 } { \operatorname { cosec } 2 x - 1 } + \frac { 1 } { \operatorname { cosec } 2 x + 1 } = \cot 2 x \sec 2 x$$ Give each answer, in degrees, to one decimal place.

6

1. The diagram shows part of the graph of \(\mathrm { y } = \operatorname { cosec } \mathrm { x }\), where \(x\) is in radians. State the equations of the three vertical asymptotes that can be seen. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-06_734_672_603_324} The tangent to the graph at the point P with \(x\)-coordinate \(\frac { \pi } { 3 }\) meets the \(x\)-axis at Q .
2. Show that the \(x\)-coordinate of Q is \(\frac { \pi } { 3 } + \sqrt { 3 }\). (You may use without proof the result that the derivative of \(\operatorname { cosec } x\) is \(- \operatorname { cosec } x \cot x\).)

14 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that the expression \(t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } + t _ { 1 } t _ { 2 } + \frac { 1 } { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the normals at the points A and B in Fig. C2.

5 Fig. 5 shows part of the curve \(y = \operatorname { cosec } x\) together with the \(x\) - and \(y\)-axes. \begin{figure}[h] \end{figure}

1. For the section of the curve which is shown in Fig. 5, write down
   1. the equations of the two vertical asymptotes,
   2. the coordinates of the minimum point.
2. Show that the equation \(x = \operatorname { cosec } x\) has a root which lies between \(x = 1\) and \(x = 2\).
3. Use the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\), with \(x _ { 0 } = 1\), to find
   1. the values of \(x _ { 1 }\) and \(x _ { 2 }\), correct to 5 decimal places,
   2. this root of the equation, correct to 3 decimal places.
4. There is another root of \(x = \operatorname { cosec } x\) which lies between \(x = 2\) and \(x = 3\). Determine whether the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\) with \(x _ { 0 } = 2.5\) converges to this root.
5. Sketch the staircase or cobweb diagram for the iteration, starting with \(x _ { 0 } = 2.5\), on the diagram in the Printed Answer Booklet.

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.

6

1. Sketch the curve with equation \(y = \operatorname { cosec } x\) for \(0 < x < \pi\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } \operatorname { cosec } x \mathrm {~d} x\), giving your answer to three significant figures.

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)

6

1. Show that $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) }$$ can be written as \(\operatorname { cosec } ^ { 2 } x\).
2. Hence solve the equation $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) } = \operatorname { cosec } x + 3$$ giving the values of \(x\) to the nearest degree in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
3. Hence solve the equation $$\frac { \sec ^ { 2 } \left( 2 \theta - 60 ^ { \circ } \right) } { \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) + 1 \right) \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) - 1 \right) } = \operatorname { cosec } \left( 2 \theta - 60 ^ { \circ } \right) + 3$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

8

1. Show that $$\int _ { 0 } ^ { \ln 2 } \mathrm { e } ^ { 1 - 2 x } \mathrm {~d} x = \frac { 3 } { 8 } \mathrm { e }$$
2. Use the substitution \(u = \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 4 } x \sqrt { \tan x } d x$$ (8 marks)

4 It is given that \(\tan ^ { 2 } x = \sec x + 11\).

1. Show that the equation \(\tan ^ { 2 } x = \sec x + 11\) can be written in the form $$\sec ^ { 2 } x - \sec x - 12 = 0$$
2. Hence show that \(\cos x = \frac { 1 } { 4 }\) or \(\cos x = - \frac { 1 } { 3 }\).
3. Hence, or otherwise, solve the equation \(\tan ^ { 2 } x = \sec x + 11\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Solve the equation \(\sec x = 5\), giving all the values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
2. Show that the equation \(\tan ^ { 2 } x = 3 \sec x + 9\) can be written as $$\sec ^ { 2 } x - 3 \sec x - 10 = 0$$
3. Solve the equation \(\tan ^ { 2 } x = 3 \sec x + 9\), giving all the values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.

7

1. Given that \(z = \frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x\).
2. Sketch the curve with equation \(y = \sec x\) for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
3. The region \(R\) is bounded by the curve \(y = \sec x\), the \(x\)-axis and the lines \(x = 0\) and \(x = 1\). Find the volume of the solid formed when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer to three significant figures.

2

1. Solve the equation \(\sec x = 3\), giving the values of \(x\) in radians to two decimal places in the interval \(0 \leqslant x < 2 \pi\).
2. Show that the equation \(\tan ^ { 2 } x = 2 \sec x + 2\) can be written as \(\sec ^ { 2 } x - 2 \sec x - 3 = 0\).
3. Solve the equation \(\tan ^ { 2 } x = 2 \sec x + 2\), giving the values of \(x\) in radians to two decimal places in the interval \(0 \leqslant x < 2 \pi\).

2

1. The diagram shows the graph of \(y = \sec x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   
   [diagram]
   1. The point \(A\) on the curve is where \(x = 0\). State the \(y\)-coordinate of \(A\).
   2. Sketch, on the axes given on page 3, the graph of \(y = | \sec 2 x |\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(\sec x = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. Solve the equation \(\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

7

1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
   [0pt] [6 marks]

9

1. It is given that \(\sec x - \tan x = - 5\).
   1. Show that \(\sec x + \tan x = - 0.2\).
   2. Hence find the exact value of \(\cos x\).
2. Hence solve the equation $$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$ giving all values of \(x\), to one decimal place, in the interval \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
   [0pt] [3 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

1. Given that

$$x = \sec ^ { 2 } y + \tan y ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos ^ { 2 } y } { 2 \tan y + 1 } .$$

7. (a) (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 \(b \sqrt { 6 }\).
(b) Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$

1. (a) Prove, by counter-example, that the statement

$$\text { "cosec } \theta - \sin \theta > 0 \text { for all values of } \theta \text { in the interval } 0 < \theta < \pi \text { " }$$ is false.
(b) Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.

2. Find, to 2 decimal places, the solutions of the equation $$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).

3. $$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5]. A more accurate estimate of this root is to be found using iterations of the form $$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
2. Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places. The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
3. Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.

1. Solve the equation

$$3 \operatorname { cosec } \theta ^ { \circ } + 8 \cos \theta ^ { \circ } = 0$$ for \(\theta\) in the interval \(0 \leq \theta \leq 180\), giving your answers to 1 decimal place.