1.05a Sine, cosine, tangent: definitions for all arguments

12 \includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm , held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are \(A , B , C , D\), \(E\) and \(F\). Points \(P\) and \(Q\) are situated where straight sections of the rope meet the pipe with centre \(A\).

1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
2. Find the length of the rope.
3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
4. Find the area of the complete region enclosed by the rope.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).

1. State the greatest and least values of \(y\).
2. Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
3. By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
   1. \(k = - 3\)
   2. \(k = 1\)
   3. \(k = 3\) Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\ & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\ & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
4. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
5. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.

1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

1 The acute angle \(x\) radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),

1. \(\tan ( \pi - x )\),
2. \(\tan \left( \frac { 1 } { 2 } \pi - x \right)\),
3. \(\sin x\).

3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).

1. Find an expression, in terms of \(k\), for
   1. \(\sin \theta\),
   2. \(\tan \theta\).
   3. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).

5 The equation of a curve is \(y = 2 \cos x\).

1. Sketch the graph of \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\), stating the coordinates of the point of intersection with the \(y\)-axis. Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac { 1 } { 3 } \pi\) and \(\pi\) respectively.
2. Find the length of \(P Q\) correct to 1 decimal place.
   The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H ( h , 0 )\) and the \(y\)-axis at \(K ( 0 , k )\).
3. Show that \(h = \frac { 5 } { 9 } \pi\) and find the value of \(k\).

10 The function f is defined by \(\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Solve the equation \(\mathrm { f } ( x ) + 4 = 0\), giving your answer correct to 1 decimal place.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
3. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).

4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).

1. Find the values of the constants \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).

7

1. 1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
   2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
   1. Find the \(x\)-coordinate of \(A\).
   2. Find the \(y\)-coordinate of \(B\).

9 The function f is defined by \(\mathrm { f } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).

1. State the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant.
3. State the largest value of \(p\) for which g has an inverse.
4. For this value of \(p\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

4 Angle \(x\) is such that \(\sin x = a + b\) and \(\cos x = a - b\), where \(a\) and \(b\) are constants.

1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(x\).
2. In the case where \(\tan x = 2\), express \(a\) in terms of \(b\).

6 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-2_526_659_1336_742} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) on \(O A\) is such that \(B C\) is perpendicular to \(O A\). The point \(D\) is on \(B C\) and the circular arc \(A D\) has centre \(C\).

1. Find \(A C\) in terms of \(r\) and \(\theta\).
2. Find the perimeter of the shaded region \(A B D\) when \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 4\), giving your answer as an exact value.

8 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-3_408_686_264_731} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius 4 cm . Angle \(A O B\) is \(\alpha\) radians. The point \(D\) on \(O B\) is such that \(A D\) is perpendicular to \(O B\). The arc \(D C\), with centre \(O\), meets \(O A\) at \(C\).

1. Find an expression in terms of \(\alpha\) for the perimeter of the shaded region \(A B D C\).
2. For the case where \(\alpha = \frac { 1 } { 6 } \pi\), find the area of the shaded region \(A B D C\), giving your answer in the form \(k \pi\), where \(k\) is a constant to be determined.

5 A curve is defined by the parametric equations $$x = 2 \tan \theta , \quad y = 3 \sin 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 4 } \theta - 3 \cos ^ { 2 } \theta\).
2. Find the coordinates of the stationary point.
3. Find the gradient of the curve at the point \(\left( 2 \sqrt { } 3 , \frac { 3 } { 2 } \sqrt { } 3 \right)\).

3

1. Solve the equation \(| 3 u + 1 | = | 2 u - 5 |\).
2. Hence solve the equation \(| 3 \cot x + 1 | = | 2 \cot x - 5 |\) for \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.

5 Use the substitution \(u = 4 - 3 \cos x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 9 \sin 2 x } { \sqrt { ( 4 - 3 \cos x ) } } \mathrm { d } x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c861e691-66da-4269-9057-4a343be9835e-12_357_565_260_790} The diagram shows the curve \(y = 5 \sin ^ { 2 } x \cos ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]

4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

4

1. Show that the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ can be written in the form $$( 3 \sqrt { } 3 ) \sin x = \cos x$$
2. Hence solve the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ for \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

4

1. Show that the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\) can be written in the form \(\tan x = k\), where \(k\) is a constant.
2. Hence solve the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{9e1bd528-e7c4-4936-a05a-dde1d1ace7c2-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 The curve with equation \(y = \frac { \sin 2 x } { \mathrm { e } ^ { 2 x } }\) has one stationary point in the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the exact \(x\)-coordinate of this point.

5 \includegraphics[max width=\textwidth, alt={}, center]{96a4df57-b3c7-4dbf-9bea-bb00ed6a4a16-2_512_775_1318_683} The diagram shows the curve \(y = \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). A rectangle \(O A B C\) is drawn, where \(B\) is the point on the curve with \(x\)-coordinate \(\theta\), and \(A\) and \(C\) are on the axes, as shown. The shaded region \(R\) is bounded by the curve and by the lines \(x = \theta\) and \(y = 0\).

1. Find the area of \(R\) in terms of \(\theta\).
2. The area of the rectangle \(O A B C\) is equal to the area of \(R\). Show that $$\theta = \frac { 1 - \sin \theta } { \cos \theta }$$
3. Use the iterative formula \(\theta _ { n + 1 } = \frac { 1 - \sin \theta _ { n } } { \cos \theta _ { n } }\), with initial value \(\theta _ { 1 } = 0.5\), to determine the value of \(\theta\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 The parametric equations of a curve are $$x = \cos 2 \theta - \cos \theta , \quad y = 4 \sin ^ { 2 } \theta$$ for \(0 \leqslant \theta \leqslant \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 \cos \theta } { 1 - 4 \cos \theta }\).
2. Find the coordinates of the point on the curve at which the gradient is - 4 .

6

1. Find
   1. \(\int \frac { \mathrm { e } ^ { 2 x } + 6 } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x\),
   2. \(\int 3 \cos ^ { 2 } x \mathrm {~d} x\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { 6 } { \ln ( x + 2 ) } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
   1. Express \(3 \cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
   2. Hence solve the equation $$3 \cos 2 x + \sin 2 x = 2$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).