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

5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.

4. (a) Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1 .$$ (c) Solve the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2 ,$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place.

2. Giving your answers to 1 decimal place, solve the equation $$5 \tan ^ { 2 } 2 \theta - 13 \sec 2 \theta = 1 ,$$ for \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).

4. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).

1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$f : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi ,$$
2. find the values of the constants \(a\) and \(b\),
3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.

6. (a) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z } .$$ (b) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7 ,$$ giving your answers to 2 decimal places.

17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).

6 In this question you must show detailed reasoning. Solve the equation \(\tan x - 3 \cot x = 2\) for values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.

6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.

4 It is given that \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\).

1. Show that the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\) can be written in the form $$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
2. Hence show that \(\tan x = 2\) or \(\tan x = - \frac { 1 } { 3 }\).
3. Hence, or otherwise, solve the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\), giving all values of \(x\) in radians to one decimal place in the interval \(- \pi < x \leqslant \pi\).

4

1. Solve the equation \(\sec x = \frac { 3 } { 2 }\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. By using a suitable trigonometrical identity, solve the equation $$2 \tan ^ { 2 } x = 10 - 5 \sec x$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Solve the equation $$\operatorname { cosec } x = 3$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (2 marks)
2. By using a suitable trigonometric identity, solve the equation $$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (6 marks)

3

1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)
2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
   1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
   2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)

7 Sketch the graph of $$y = \cot \left( x - \frac { \pi } { 2 } \right)$$ for \(0 \leq x \leq 2 \pi\) [0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-08_1650_1226_587_408} \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-09_2488_1716_219_153}

15

1. Given that $$y = \operatorname { cosec } \theta$$ 15
   1. (i) Express \(y\) in terms of \(\sin \theta\). 15
   2. (ii) Hence, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$ 15
   3. (iii) Show that $$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$ 15
   4. 1. Use the substitution $$x = 2 \operatorname { cosec } u$$ to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x \quad \text { for } x > 2$$ can be written as $$k \int \sin u \mathrm {~d} u$$ where \(k\) is a constant to be found.
         15
   5. (ii) Hence, show $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 4 } } { 4 x } + c \quad \text { for } x > 2$$ where \(c\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-32_2492_1721_217_150}
      \includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-36_2496_1721_214_148}

4

1. Sketch the graph of \(y = \sec \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve \(\sec \theta = \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).

10

1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$

Solve the equation $$6 \sec ^ { 2 } x - 8 = \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]

By using appropriate trigonometrical identities, find the exact value of

1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]

Solve the equation \(\sec^2 \theta = 3 \cosec \theta\) for \(0° < \theta < 180°\). [5]