1.05g Exact trigonometric values: for standard angles

2

1. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_582_922_335_575} The diagram shows the curve \(y = k \cos \left( x - \frac { 1 } { 6 } \pi \right)\) where \(k\) is a positive constant and \(x\) is measured in radians. The curve crosses the \(x\)-axis at point \(A\) and \(B\) is a minimum point. Find the coordinates of \(A\) and \(B\).
2. Find the exact value of \(t\) that satisfies the equation $$3 \sin ^ { - 1 } ( 3 t ) + 2 \cos ^ { - 1 } \left( \frac { 1 } { 2 } \sqrt { 2 } \right) = \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-04_2718_33_141_2013}

5 \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).

1. Find the values of the positive integers \(a , b\) and \(c\).
2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
   1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
   2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\) The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .

5 The first, third and fifth terms of an arithmetic progression are \(2 \cos x , - 6 \sqrt { 3 } \sin x\) and \(10 \cos x\) respectively, where \(\frac { 1 } { 2 } \pi < x < \pi\).

1. Find the exact value of \(x\).
2. Hence find the exact sum of the first 25 terms of the progression.

6 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).

1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).

1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 12 cm . The lines \(A X\) and \(B X\) are tangents to the circle at \(A\) and \(B\) respectively. Angle \(A O B = \frac { 1 } { 3 } \pi\) radians.

1. Find the exact length of \(A X\), giving your answer in terms of \(\sqrt { } 3\).
2. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).

3 The function \(\mathrm { f } : x \mapsto a + b \cos x\) is defined for \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } ( 0 ) = 10\) and that \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1\), find

1. the values of \(a\) and \(b\),
2. the range of \(f\),
3. the exact value of \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).

3 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_584_659_575_742} In the diagram, \(A B C\) is an equilateral triangle of side 2 cm . The mid-point of \(B C\) is \(Q\). An arc of a circle with centre \(A\) touches \(B C\) at \(Q\), and meets \(A B\) at \(P\) and \(A C\) at \(R\). Find the total area of the shaded regions, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }

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 \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).

3 In the diagram, \(A C\) is an arc of a circle, centre \(O\) and radius 6 cm . The line \(B C\) is perpendicular to \(O C\) and \(O A B\) is a straight line. Angle \(A O C = \frac { 1 } { 3 } \pi\) radians. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

3 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772} In the diagram, \(A O B\) is a sector of a circle with centre \(O\) and radius 12 cm . The point \(A\) lies on the side \(C D\) of the rectangle \(O C D B\). Angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express the area of the shaded region in the form \(a ( \sqrt { } 3 ) - b \pi\), stating the values of the integers \(a\) and \(b\).

9 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_554_723_1557_712} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 0 , - 2 )\) and \(C\) is \(( 12,14 )\). The diagonal \(B D\) is parallel to the \(x\)-axis.

1. Explain why the \(y\)-coordinate of \(D\) is 6 . The \(x\)-coordinate of \(D\) is \(h\).
2. Express the gradients of \(A D\) and \(C D\) in terms of \(h\).
3. Calculate the \(x\)-coordinates of \(D\) and \(B\).
4. Calculate the area of the rectangle \(A B C D\).

3

1. Sketch, on a single diagram, the graphs of \(y = \cos 2 \theta\) and \(y = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Write down the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(0 \leqslant \theta \leqslant 2 \pi\).
3. Deduce the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(10 \pi \leqslant \theta \leqslant 20 \pi\).

5

1. Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Verify that \(x = 30 ^ { \circ }\) is a root of the equation \(\sin x = \cos 2 x\), and state the other root of this equation for which \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
3. Hence state the set of values of \(x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), for which \(\sin x < \cos 2 x\).

6

1. Show that \(4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta\).
2. Find the exact value of \(4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi\).
3. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x\).

6 It is given that \(3 \sin 2 \theta = \cos \theta\) where \(\theta\) is an angle such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

1. Find the exact value of \(\sin \theta\).
2. Find the exact value of \(\sec \theta\).
3. Find the exact value of \(\cos 2 \theta\).

4

1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$3 \sin \theta + 4 \cos \theta = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), correct to 1 decimal place.
3. Write down the least value of \(3 \sin \theta + 4 \cos \theta + 7\) as \(\theta\) varies.

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 \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

3

1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-05_446_851_260_646} A block of mass \(m \mathrm {~kg}\) is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). The other string is inclined at \(60 ^ { \circ }\) to the horizontal and the tension in this string is 20 N . Find \(T\) and \(m\).

6 A block of mass 5 kg is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is \(\mu\).

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-10_424_709_392_760} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure} When a force of magnitude 40 N is applied to the block, acting up the plane parallel to a line of greatest slope, the block begins to slide up the plane (see Fig. 6.1). Show that \(\mu < \frac { 1 } { 5 } \sqrt { 3 }\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-11_422_727_264_749} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} When a force of magnitude 40 N is applied horizontally, in a vertical plane containing a line of greatest slope, the block does not move (see Fig. 6.2). Show that, correct to 3 decimal places, the least possible value of \(\mu\) is 0.152 .

3 A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal table. The string makes an angle of \(24 ^ { \circ }\) above the horizontal and the tension in the string is 30 N .

1. Draw a diagram showing all the forces acting on the block.
2. Find the coefficient of friction between the block and the table.

3 \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-04_519_616_260_762} Coplanar forces of magnitudes \(24 \mathrm {~N} , P \mathrm {~N} , 20 \mathrm {~N}\) and 36 N act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the values of \(P\) and \(\theta\).