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

6. (a) Show that $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv 1 - \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } + 2 = 0$$ Give your answers in terms of \(\pi\).

1. (i) Showing each step in your reasoning, prove that

$$( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$3 \sin \theta = \tan \theta$$ giving your answers in degrees to 1 decimal place, as appropriate.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8. In this question the angle \(\theta\) is measured in degrees throughout.

1. Show that the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta , \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ may be rewritten as $$6 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta$$ Give your answers to one decimal place, where appropriate.

9. (i) Find the exact value of \(x\) for which $$2 \log _ { 10 } ( x - 2 ) - \log _ { 10 } ( x + 5 ) = 0$$ (ii) Given $$\log _ { p } ( 4 y + 1 ) - \log _ { p } ( 2 y - 2 ) = 1 \quad p > 2 , y > 1$$ express \(y\) in terms of \(p\).

13. (a) Show that the equation $$5 \cos x + 1 = \sin x \tan x$$ can be written in the form $$6 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 \cos 2 \theta + 1 = \sin 2 \theta \tan 2 \theta$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]

1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.

12. (a) Show that $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ may be expressed in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ giving your answers in radians to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}

10. (a) Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}

12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.

12. (a) Show that the equation $$6 \cos x - 5 \tan x = 0$$ may be expressed in the form $$6 \sin ^ { 2 } x + 5 \sin x - 6 = 0$$ (b) Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$6 \cos \left( 2 \theta - 10 ^ { \circ } \right) - 5 \tan \left( 2 \theta - 10 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (a) Show that the equation $$8 \tan \theta = 3 \cos \theta$$ may be rewritten in the form $$3 \sin ^ { 2 } \theta + 8 \sin \theta - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x \leqslant 90 ^ { \circ }\), the equation $$8 \tan 2 x = 3 \cos 2 x$$ giving your answers to 2 decimal places.

1. (a) Show that the equation

$$\frac { 3 \sin \theta \cos \theta } { 2 \sin \theta - 1 } = 5 \tan \theta \quad \sin \theta \neq \frac { 1 } { 2 }$$ can be written in the form $$3 \sin ^ { 3 } \theta + 10 \sin ^ { 2 } \theta - 8 \sin \theta = 0$$ (b) Hence solve, for \(- \frac { \pi } { 4 } < x < \frac { \pi } { 4 }\) $$\frac { 3 \sin 2 x \cos 2 x } { 2 \sin 2 x - 1 } = 5 \tan 2 x$$ giving your answers to 3 decimal places where appropriate.

7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation $$3 \sin \left( 2 x - 15 ^ { \circ } \right) = \cos \left( 2 x - 15 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. Solve, for \(0 < \theta < 2 \pi\), the equation $$4 \sin ^ { 2 } \theta + 8 \cos \theta = 3$$ giving your answers to 3 significant figures.

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

Solutions based entirely on calculator technology are not acceptable.

1. Solve, for \(- \frac { \pi } { 2 } < x < \pi\), the equation $$5 \sin ( 3 x + 0.1 ) + 2 = 0$$ giving your answers, in radians, to 2 decimal places.
2. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$2 \tan \theta \sin \theta = 5 + \cos \theta$$ giving your answers, in degrees, to one decimal place.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0e08d931-aa1c-48a8-8b39-47096f981950-26_643_736_721_660} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
   The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
   Given that the \(y\) coordinate of \(P\) is 7
   (a) state the value of \(A\),
   (b) find the exact coordinates of \(Q\),
   (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.

9. (a) Show that the equation $$\cos \theta - 1 = 4 \sin \theta \tan \theta$$ can be written in the form $$5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0$$ (b) Hence solve, for \(0 \leqslant x < \frac { \pi } { 2 }\) $$\cos 2 x - 1 = 4 \sin 2 x \tan 2 x$$ giving your answers, where appropriate, to 2 decimal places.

8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. (a) Show that the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ can be written in the form $$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$ where \(a , b\) and \(c\) are constants to be found.
   (b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-27_2644_1840_118_111}

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$3 \cos \theta ( \tan \theta \sin \theta + 3 ) = 11 - 5 \cos \theta$$ may be written as $$3 \cos ^ { 2 } \theta - 14 \cos \theta + 8 = 0$$
2. Hence solve, for \(0 < x < 360 ^ { \circ }\) $$3 \cos 2 x ( \tan 2 x \sin 2 x + 3 ) = 11 - 5 \cos 2 x$$ giving your answers to one decimal place.

9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.

1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$ giving your answers to one decimal place.
2. The first three terms of an arithmetic sequence are $$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$ where \(\alpha\) is a constant.
   1. Show that \(2 \cos \alpha = 3 \sin ^ { 2 } \alpha\) Given that \(\pi < \alpha < 2 \pi\),
   2. find, showing all working, the value of \(\alpha\) to 3 decimal places.

7. (i) Show that $$\tan \theta + \frac { 1 } { \tan \theta } \equiv \frac { 1 } { \sin \theta \cos \theta } \quad \theta \neq \frac { \mathrm { n } \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) Solve, for \(0 \leqslant x < 90 ^ { \circ }\), the equation $$3 \cos ^ { 2 } \left( 2 x + 10 ^ { \circ } \right) = 1$$ giving your answers in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) the equation $$2 \tan \theta + 3 \sin \theta = 0$$ giving your answers, as appropriate, to one decimal place.
2. Hence, or otherwise, find the smallest positive solution of $$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$ giving your answer to one decimal place.

9. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\) (ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$

1. find \(\cos x\) in terms of \(k\)
2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
   \includegraphics[max width=\textwidth, alt={}]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-30_2671_1942_107_121}