1.05g Exact trigonometric values: for standard angles

7 Fig. 7 shows the graphs of the curves \(y = \mathrm { e } ^ { - x }\) and \(y = \mathrm { e } ^ { - x } \sin x\) for \(0 \leq x \leq \pi\). \begin{figure}[h] \end{figure} The maximum point on \(y = \mathrm { e } ^ { - x } \sin x\) is at A , and the curves touch at B . \(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) are the points on the \(x\)-axis such that \(\mathrm { A } ^ { \prime } \mathrm { A }\) and \(\mathrm { B } ^ { \prime } \mathrm { B }\) are parallel to the \(y\)-axis.
Show that \(\mathrm { OA } ^ { \prime } = \mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).

4 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 1 + \cos x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
P is the point on the curve with \(x\)-coordinate \(\frac { 1 } { 3 } \pi\). \begin{figure}[h] \end{figure}

1. Find the \(y\)-coordinate of P .
2. Find \(\mathrm { f } ^ { \prime } ( x )\). Hence find the gradient of the curve at the point P .
3. Show that the derivative of \(\frac { \sin x } { 1 + \cos x }\) is \(\frac { 1 } { 1 + \cos x }\). Hence find the exact area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 3 } \pi\).
4. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arccos \left( \frac { 1 } { x } - 1 \right)\). State the domain of this inverse function, and add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.

7 A curve is defined by the equation \(\sin 2 x + \cos y = \sqrt { 3 }\).

1. Verify that the point \(\mathrm { P } \left( \frac { 1 } { 6 } \pi , \frac { 1 } { 6 } \pi \right)\) lies on the curve.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P .

3 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).

1. Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre \(\mathrm { O } . \mathrm { AB }\) is one of the sides of the polygon. C is the midpoint of AB . Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_457_422_457_936} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{figure} (A) Show that \(\mathrm { AB } = 2 \sin 15 ^ { \circ }\).
   (B) Use a double angle formula to express \(\cos 30 ^ { \circ }\) in terms of \(\sin 15 ^ { \circ }\). Using the exact value of \(\cos 30 ^ { \circ }\), show that \(\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }\).
   (C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6 \sqrt { 2 - \sqrt { 3 } }\).
2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c0fcd64b-8ca0-4309-9f58-c23cc4208f4d-2_450_416_1562_940} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} (A) Show that \(\mathrm { DE } = 2 \tan 15 ^ { \circ }\).
   (B) Let \(t = \tan 15 ^ { \circ }\). Use a double angle formula to express \(\tan 30 ^ { \circ }\) in terms of \(t\). Hence show that \(t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0\).
   (C) Solve this equation, and hence show that \(\pi < 12 ( 2 - \sqrt { 3 } )\).
3. Use the results in parts (i)( \(C\) ) and (ii)( \(C\) ) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form.

3 Using appropriate right-angled triangles, show that \(\tan 45 ^ { \circ } = 1\) and \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Hence show that \(\tan 75 ^ { \circ } = 2 + \sqrt { 3 }\).

1.(a)Write down the exact value of \(\cos 405 ^ { \circ }\) (b)Hence,using a double angle identity for cosine,or otherwise,determine the exact value of \(\cos 101.25 ^ { \circ }\) ,giving your answer in the form $$a \sqrt { b + c \sqrt { 2 + \sqrt { 2 } } }$$ where \(a\) ,\(b\) and \(c\) are rational numbers.

3. \begin{figure}[h] \end{figure} Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively. Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).

1. Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
2. Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
3. Hence find the value of \(k\) By considering triangle \(D B C\), or otherwise,
4. find the exact value of \(\sin 54 ^ { \circ }\)

4 Sandy is throwing a stone at a plum tree. The stone is thrown from a point O at a speed of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\). You are given that, \(t\) seconds after being thrown, the stone is \(\left( 9.8 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) higher than O . When descending, the stone hits a plum which is 3.675 m higher than O . Air resistance should be neglected. Calculate the horizontal distance of the plum from O .

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = a \\ & C _ { 2 } : r = 2 a | \cos \theta | \end{aligned}$$ where \(a\) is a positive constant. The curves intersect at the points \(P _ { 1 }\) and \(P _ { 2 }\).

1. Find the polar coordinates of \(P _ { 1 }\) and \(P _ { 2 }\).
2. In a single diagram, sketch \(C _ { 1 } , C _ { 2 }\) and their line of symmetry.
3. The region \(R\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is bounded by the \(\operatorname { arcs } O P _ { 1 } , P _ { 1 } P _ { 2 }\) and \(P _ { 2 } O\), where \(O\) is the pole. Find the area of \(R\), giving your answer in exact form.

4 \includegraphics[max width=\textwidth, alt={}, center]{ae8d874a-5c1d-45bb-b853-d12006004b7f-2_519_583_1384_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).

1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
2. Find the speed of the bead as it leaves the wire at \(B\).
3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).

2 A particle \(P\) moves on a straight line \(A O B\) in simple harmonic motion. The centre of the motion is \(O\), and \(P\) is instantaneously at rest at \(A\) and \(B\). The point \(C\) is on the line \(A O B\), between \(A\) and \(O\), and \(C O = 10 \mathrm {~m}\). When \(P\) is at \(C\), the magnitude of its acceleration is \(0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and it is moving towards \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the period of the motion, in terms of \(\pi\),
2. the amplitude of the motion. The point \(M\) is the mid-point of \(O B\). Find the time that \(P\) takes to travel directly from \(C\) to \(M\).

4 \includegraphics[max width=\textwidth, alt={}, center]{baea9836-ea05-442f-9e87-a2a1480dc74c-2_338_957_1482_593} A uniform rod \(B C\) of length \(2 a\) and weight \(W\) is hinged to a fixed point at \(B\). A particle of weight \(3 W\) is attached to the rod at \(C\). The system is held in equilibrium by a light elastic string of natural length \(\frac { 3 } { 5 } a\) in the same vertical plane as the rod. One end of the elastic string is attached to the rod at \(C\) and the other end is attached to a fixed point \(A\) which is at the same horizontal level as \(B\). The rod and the string each make an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). Find

1. the modulus of elasticity of the string,
2. the magnitude and direction of the force acting on the rod at \(B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{eb3dccaf-d151-472d-82f3-6ba215b0b7f0-2_339_957_1482_593} A uniform rod \(B C\) of length \(2 a\) and weight \(W\) is hinged to a fixed point at \(B\). A particle of weight \(3 W\) is attached to the rod at \(C\). The system is held in equilibrium by a light elastic string of natural length \(\frac { 3 } { 5 } a\) in the same vertical plane as the rod. One end of the elastic string is attached to the rod at \(C\) and the other end is attached to a fixed point \(A\) which is at the same horizontal level as \(B\). The rod and the string each make an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). Find

1. the modulus of elasticity of the string,
2. the magnitude and direction of the force acting on the rod at \(B\).

3 A particle moves on a straight line \(A O B\) in simple harmonic motion, where \(A B = 2 a \mathrm {~m}\). The centre of the motion is \(O\) and the particle is instantaneously at rest at \(A\) and \(B\). The point \(M\) is the mid-point of \(O B\). The particle passes through \(M\) moving towards \(O\) and next achieves its maximum speed one second later. Find the period of the motion. Find the distance of the particle from \(O\) when its speed is equal to one half of its maximum speed. At an instant 2.5 seconds after the particle passes through \(M\) moving towards \(O\), the distance of the particle from \(O\) is \(\sqrt { } 2 \mathrm {~m}\). Find, in metres, the amplitude of the motion.

2 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

2 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

2 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

5

1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.

2 The height of the first part of a rollercoaster track is \(h \mathrm {~m}\) at a horizontal distance of \(x \mathrm {~m}\) from the start. A student models this using the equation \(h = 17 + 15 \cos 6 x\), for \(0 \leqslant x \leqslant 40\), using the values of \(h\) given when their calculator is set to work in degrees.

1. Find the height that the student's model predicts when the horizontal distance from the start is 40 m .
2. The student argues that the model predicts that the rollercoaster track will achieve a maximum height of 32 m more than once because the cosine function is periodic. Comment on the validity of the student's argument.

3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h] \end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).

1. (a) Write down the exact value of \(\cos \frac { \pi } { 6 }\).

The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(c) Using your answer to part (b), find an estimate for the area of \(S\).

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 $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$