1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance \(y\) metres from the line TOA. Other distances and angles are as shown. \begin{figure}[h] \end{figure}

1. Show that \(\theta = \beta - \alpha\), and hence that \(\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }\). Calculate the angle \(\theta\) when \(y = 6\).
2. By differentiating implicitly, show that \(\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta\).
3. Use this result to find the value of \(y\) that maximises the angle \(\theta\). Calculate this maximum value of \(\theta\). [You need not verify that this value is indeed a maximum.]

4 Given that \(x = 2 \sec \theta\) and \(y = 3 \tan \theta\), show that \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1\).

6

1. 1. By applying De Moivre's theorem to \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { 3 }\), show that $$\cos 3 \theta = \cos ^ { 3 } \theta - 3 \cos \theta \sin ^ { 2 } \theta$$
   2. Find a similar expression for \(\sin 3 \theta\).
   3. Deduce that $$\tan 3 \theta = \frac { \tan ^ { 3 } \theta - 3 \tan \theta } { 3 \tan ^ { 2 } \theta - 1 }$$
2. 1. Hence show that \(\tan \frac { \pi } { 12 }\) is a root of the cubic equation $$x ^ { 3 } - 3 x ^ { 2 } - 3 x + 1 = 0$$
   2. Find two other values of \(\theta\), where \(0 < \theta < \pi\), for which \(\tan \theta\) is a root of this cubic equation.
3. Hence show that $$\tan \frac { \pi } { 12 } + \tan \frac { 5 \pi } { 12 } = 4$$

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).

3 In this question you must show detailed reasoning. Given that \(5 \sin 2 x = 3 \cos x\), where \(0 ^ { \circ } < x < 90 ^ { \circ }\), find the exact value of \(\sin x\).

3 It is given that \(\sin \theta = - 0.1\) and \(180 ^ { \circ } < \theta < 270 ^ { \circ }\) Find the exact value of \(\cos \theta\)

12

1. Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\) 12
2. Factorise \(\mathrm { p } ( x )\) completely.
   12
3. Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

8

1. Given that $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ show that $$8 \cot ^ { 2 } \theta - 2 \cot \theta - 1 = 0$$ 8
2. Hence, solve $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ in the interval \(0 < \theta < 2 \pi\) Give your answers to two decimal places.
   8
3. Solve $$9 \sin ^ { 2 } \left( 2 x - \frac { \pi } { 4 } \right) + \sin \left( 4 x - \frac { \pi } { 2 } \right) = 8$$ in the interval \(0 < x < \frac { \pi } { 2 }\) Give your answers to one decimal place.

10

1. Point \(A\) on the curve has coordinates ( \(a , 0.5\) )
   10
   1. (i) Find the value of \(a\) 10
   2. (ii) State the value of \(\sin \left( 180 ^ { \circ } - a ^ { \circ } \right)\) 10
   3. Point \(B\) on the curve has coordinates \(\left( b , - \frac { 3 } { 7 } \right)\) 10
   4. 1. Find the exact value of \(\sin \left( b ^ { \circ } - 180 ^ { \circ } \right)\) 10
   5. (ii) Find the exact value of \(\cos b ^ { \circ }\)

9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design. The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2. \begin{figure}[h] \end{figure} 9

1. Show that the \(x\)-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
2. Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
3. It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\) 9
4. Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
   \end{figure} Find, in terms of \(d\), the exact distance \(O Q\). \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

8

1. Given that \(\cos \theta \neq \pm 1\), prove the identity $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } \equiv 2 \operatorname { cosec } ^ { 2 } \theta$$ 8
2. Hence, find the set of values of \(A\) for which the equation $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = A$$ has real solutions.
   Fully justify your answer.
   8
3. Given that \(\theta\) is obtuse and $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = 16$$ find the exact value of \(\cot \theta\)

7 \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-10_854_1027_264_520} The diagram shows a curve known as an astroid.
The curve has parametric equations $$\begin{aligned} & x = 4 \cos ^ { 3 } t \\ & y = 4 \sin ^ { 3 } t \\ & ( 0 \leq t < 2 \pi ) \end{aligned}$$ The section of the curve from \(t = 0\) to \(t = \frac { \pi } { 2 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the curved surface area of the shape formed is equal to \(\frac { b \pi } { c }\), where \(b\) and \(c\) are integers.

10 Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$ Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
[0pt] [Question 11 is printed on the next page.]

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta = 3 \cos \theta$$ can be written as $$3 \sin ^ { 2 } \theta + 2 \sin \theta - 3 = 0$$
2. Hence solve, for \(- \pi < x < \pi\), the equation $$2 \tan \left( 2 x + \frac { \pi } { 3 } \right) = 3 \cos \left( 2 x + \frac { \pi } { 3 } \right)$$ giving your answers to 3 significant figures.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$3 \tan ^ { 2 } \theta + 7 \sec \theta - 3 = 0$$ giving your answers to one decimal place.

3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.

1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.

12 A projectile is launched from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right)$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

10

1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

9

1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).

6 Given that the angle \(\theta\) is acute and \(\cos \theta = \frac { 3 } { 4 }\) find, without using a calculator, the exact value of \(\sin 2 \theta\) and of \(\cot \theta\).

12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.

1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right) .$$
2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 } .$$
3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.

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 \(0 \leqslant \theta \leqslant 2 \pi\).

8

1. Using the quotient rule, show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tan 3 \theta ) = 3 + 3 \tan ^ { 2 } 3 \theta\) for \(- \frac { 1 } { 6 } \pi < \theta < \frac { 1 } { 6 } \pi\).
2. Hence find the value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 9 } \pi } \tan ^ { 2 } 3 \theta \mathrm {~d} \theta\), giving your answer in the simplest exact form.

10

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