1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs

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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}

6 It is given that \(\alpha = \cos ^ { - 1 } \left( \frac { 8 } { 17 } \right)\).
Find, without using the trigonometric functions on your calculator, the exact value of \(\frac { 1 } { \sin \alpha } + \frac { 1 } { \tan \alpha }\).

7 A function f is defined by f : \(x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }\), where \(A\) is a constant.
3. State the largest value of \(A\) for which g has an inverse.
4. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

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 \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).

11 The function f is defined by \(\mathrm { f } : x \mapsto 4 \sin x - 1\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. State the range of f .
2. Find the coordinates of the points at which the curve \(y = \mathrm { f } ( x )\) intersects the coordinate axes.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), stating both the domain and range of \(\mathrm { f } ^ { - 1 }\). {www.cie.org.uk} after the live examination series. }

5 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),

1. find \(A M\) in terms of \(x\),
2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).

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1. Solve the equation \(\sin ^ { - 1 } ( 3 x ) = - \frac { 1 } { 3 } \pi\), giving the solution in an exact form.
2. Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leqslant \theta \leqslant \pi\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-3_602_570_260_790} In the diagram, triangle \(A B C\) is right-angled and \(D\) is the mid-point of \(B C\). Angle \(D A C = 30 ^ { \circ }\) and angle \(B A D = x ^ { \circ }\). Denoting the length of \(A D\) by \(l\),

1. express each of \(A C\) and \(B C\) exactly in terms of \(l\), and show that \(A B = \frac { 1 } { 2 } l \sqrt { } 7\),
2. show that \(x = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { } 3 } \right) - 30\).

2 Given that \(x = \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)\), find the exact value of

1. \(\cos ^ { 2 } x\),
2. \(\tan ^ { 2 } x\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-3_801_1273_255_434} The diagram shows a circle \(C _ { 1 }\) touching a circle \(C _ { 2 }\) at a point \(X\). Circle \(C _ { 1 }\) has centre \(A\) and radius 6 cm , and circle \(C _ { 2 }\) has centre \(B\) and radius 10 cm . Points \(D\) and \(E\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and \(D E\) is parallel to \(A B\). Angle \(D A X = \frac { 1 } { 3 } \pi\) radians and angle \(E B X = \theta\) radians.

1. By considering the perpendicular distances of \(D\) and \(E\) from \(A B\), show that the exact value of \(\theta\) is \(\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)\).
2. Find the perimeter of the shaded region, correct to 4 significant figures.

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1. Find the possible values of \(x\) for which \(\sin ^ { - 1 } \left( x ^ { 2 } - 1 \right) = \frac { 1 } { 3 } \pi\), giving your answers correct to 3 decimal places.
2. Solve the equation \(\sin \left( 2 \theta + \frac { 1 } { 3 } \pi \right) = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant \pi\), giving \(\theta\) in terms of \(\pi\) in your answers.

2 Find the value of \(x\) satisfying the equation \(\sin ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } ( 3 )\).

3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).

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1. Show that the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) can be expressed as $$3 \cos ^ { 2 } \theta - 4 \cos \theta - 4 = 0$$ and hence solve the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-3_581_773_1400_721} The diagram shows part of the graph of \(y = a \cos x - b\), where \(a\) and \(b\) are constants. The graph crosses the \(x\)-axis at the point \(C \left( \cos ^ { - 1 } c , 0 \right)\) and the \(y\)-axis at the point \(D ( 0 , d )\). Find \(c\) and \(d\) in terms of \(a\) and \(b\).

3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).

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1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - x$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 1.0 and 1.2.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$ with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } x = \frac { 1 } { 2 } x + 1$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify, by calculation, that this root lies between 0.5 and 1 .
3. Show that this root also satisfies the equation $$x = \sin ^ { - 1 } \left( \frac { 2 } { x + 2 } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 2 } { x _ { n } + 2 } \right)$$ with initial value \(x _ { 1 } = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

2 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).

1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. By sketching suitable graphs, show that the equation $$\sec x = 3 - x ^ { 2 }$$ has exactly one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } ^ { 2 } } \right)$$ converges, then it converges to a root of the equation given in part (i).
3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to determine the root in the interval \(0 < x < \frac { 1 } { 2 } \pi\) correct to 2 decimal places, showing the result of each iteration.

5

1. By sketching a suitable pair of graphs, show that the equation $$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$ where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 1 and 1.4.
3. Show that this root also satisfies the equation $$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
4. Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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1. By sketching a suitable pair of graphs, show that the equation $$\operatorname { cosec } \frac { 1 } { 2 } x = \frac { 1 } { 3 } x + 1$$ has one root in the interval \(0 < x \leqslant \pi\).
2. Show by calculation that this root lies between 1.4 and 1.6.
3. Show that, if a sequence of values in the interval \(0 < x \leqslant \pi\) given by the iterative formula $$x _ { n + 1 } = 2 \sin ^ { - 1 } \left( \frac { 3 } { x _ { n } + 3 } \right)$$ converges, then it converges to the root of the equation in part (i).
4. Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
3. Hence find the value of \(k\) correct to 2 decimal places.

9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).

1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
4. Find the exact value of the \(x\)-coordinate of \(M\).
5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.

6

1. By sketching a suitable pair of graphs, show that there is only one value of \(x\) in the interval \(0 < x < \frac { 1 } { 2 } \pi\) that is a root of the equation $$\cot x = x$$
2. Verify by calculation that this root lies between 0.8 and 0.9 radians.
3. Show that this value of \(x\) is also a root of the equation $$x = \tan ^ { - 1 } \left( \frac { 1 } { x } \right)$$
4. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { x _ { n } } \right)$$ to determine this root correct to 2 decimal places, showing the result of each iteration.

4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).