$\def\D{\displaystyle}\def\cosec{\text{cosec }}$
1 (CIE 2012, s, paper 11, question 6)
(i) Given that $\D 15\cos^2\theta + 2\sin^2 \theta = 7,$ show that $\D \tan^2 \theta = \frac 85
.$ [4]
(ii) Solve $\D 15\cos^2 \theta + 2\sin^2\theta = 7$ for $\D 0 \le \theta\le\pi $ radians. [3]
2 (CIE 2012, s, paper 12, question 3)
Show that $\D \cot A + \frac{\sin A}
{1 + \cos A} = \cosec A .$ [4]
3 (CIE 2012, s, paper 22, question 9)
(a) Solve the equation
(i) $\D 3 \sin x – 5 \cos x = 0$ for $\D 0^{\circ } < x < 360^{\circ},$ [3]
(ii) $\D 5 \sin^2 y + 9 \cos y - 3 = 0$ for $\D 0^{\circ} < y < 360^{\circ}.$ [5]
(b) Solve $\D \sin(3 – z) = 0.8$ for $0 < z < \pi$ radians. [4]
4 (CIE 2012, w, paper 11, question 11)
(a) Solve $\D \cosec\left(2x-\frac{\pi}{3}\right)=\sqrt{2}$ for $\D 0<x<\pi$ radians.[4]
(b)(i) Given that $\D 5(\cos y+\sin y)(2\cos y-\sin y)=7,$ show that $\D 12\tan^2y-5\tan y-3=0$[4]
(ii) Hence solve $\D 5(\cos y+\sin y)(2\cos y-\sin y)=7,$ for $\D 0^\circ<x,180^\circ.$[3]
5 (CIE 2012, w, paper 12, question 3)
(i) Show that $\D \cot\theta +\frac{\sin\theta}{1+\cos\theta}=\cosec\theta$.[5]
(ii) Explain why the equation $\D \cot\theta +\frac{\sin\theta}{1+\cos\theta}=\frac{1}{2}$ has no solution.[1]
6 (CIE 2012, w, paper 13, question 10)
(i) Solve $\D \tan^2 x-2\sec x+1=0$ for $\D 0\le x\le 360^\circ.$[4]
(ii) Solve $\D \cos^2 3y=5\sin^2 3y$ for $\D 0\le y\le 2$ radians.[4]
(iii) Solve $\D 2\cosec \left(z+\frac{\pi}{2}\right)=5$ for $\D 0\le z\le 6$ radians.[4]
7 (CIE 2012, w, paper 22, question 11)
(a) Solve $\D 4\sin x+9\cos x=0$ for $\D 0\le x\le 360^\circ$. [3]
(b) Solve $\D \cosec y-1=12\sin y$ for $\D 0<y<360^\circ$.[5]
(c) Solve $\D 3\sec\left(\frac{z}{3}\right)=5$ for $\D 0<z<6\pi $ radians.[4]
8 (CIE 2012, w, paper 23, question 6)
(a) Given that $\D \cos x=p,$ find an expression, in terms of $\D p$, for $\D \tan^2x.$
[3]
(b) Prove that $\D (\cot\theta +\tan\theta)^2=\sec^2\theta+\cosec^2\theta.$
[3]
9 (CIE 2013, s, paper 11, question 11)
(a) Solve $\D 2\sin\left(x+\frac{\pi}{3}\right)=-1$ for $\D 0\le x\le 2\pi.$
[4]
(b) Solve $\D \tan y-2=\cot y$ for $\D 0\le y\le 180^\circ.$
[6]
10 (CIE 2013, s, paper 12, question 3)
Show that $\D (1-\cos\theta-\sin\theta)^2$ $\D -2(1-\sin\theta)(1-\cos\theta)=0.$[3]
11 (CIE 2013, s, paper 12, question 11)
(a) Solve $\D \cos 2x+2\sec 2x+3=0$ for $\D 0\le x\le 360^\circ.$
[5]
(b) Solve $\D 2\sin^2\left(y-\frac{\pi}{6}\right)=1$ for $\D 0\le y\le\pi.$
[4]
12 (CIE 2013, s, paper 21, question 1)
Prove that $\D \left(\frac{1+\sin\theta}{\cos\theta}\right)^2 +
\left(\frac{1-\sin\theta}{\cos\theta}\right)^2 =2+4\tan^2\theta.$
[4]
13 (CIE 2013, w, paper 11, question 3)
Show that $\D \frac{1+\sin\theta}{\cos\theta} +
\frac{\cos\theta}{1+\sin\theta} =2\sec\theta.$
[4]
14 (CIE 2013, w, paper 13, question 3)
Show that $\D \tan^2\theta-\sin^2\theta=\sin^4\theta\sec^2\theta.$[4]
15 (CIE 2013, w, paper 13, question 9)
(a)(i) Solve $\D 6\sin^2x=5+\cos x$ for $\D 0<x<180^\circ.$[4]
(ii) Hence, or otherwise, solve $\D 6\cos^2y=5+\sin y$ for $\D 0<y<180^\circ.$[3]
(b) Solve $\D 4\cot^2z-3\cot z=0$ for $\D 0<z<\pi$ radians.[4]
16 (CIE 2013, w, paper 21, question 12)
(a) Solve the equation $\D 2\cosec x+\frac{7}{\cos x=0}$ for $\D 0\le x\le 360^\circ.$[4]
(b) Solve the equation $\D 7\sin(2y-1=5)$ for $\D 0\le y\le5$ radians.[5]
17 (CIE 2014, s, paper 11, question 1)
Show that $\D \tan\theta+\frac{\cos\theta}{1+\sin\theta}=\sec\theta.$[4]
18 (CIE 2014, s, paper 11, question 11)
(a) Solve $\D 5\sin2x+3\cos2x =0$ for $\D 0\le x\le 180^\circ.$[4]
(b) Solve $\D 2\cot^2y+3\cosec y=0$ for $\D 0\le y\le 360^\circ.$[4]
(c) Solve $\D 3\cos(z+1.2)=2$ for $\D 0\le z\le 6 $ radians.[4]
19 (CIE 2014, s, paper 12, question 1)
Show that $\D \frac{\cos A}{1+\sin a}+\frac{1+\sin A}{\cos A}$ can be written in the form $\D p\sec A$, where $\D p$ is an integer to be found.
20 (CIE 2014, s, paper 12, question 11)
(a) Solve $\D \tan^2x+5\tan x =0$ for $\D 0\le x\le 180^\circ.$[4]
(b) Solve $\D 2\cos^2y-\sin y-1=0$ for $\D 0\le y\le 360^\circ.$[4]
(c) Solve $\D \sec(2z-\frac{\pi}{6})$ for $\D 0\le z\le \pi $ radians.[4]
1 (CIE 2012, s, paper 11, question 6)
(i) Given that $\D 15\cos^2\theta + 2\sin^2 \theta = 7,$ show that $\D \tan^2 \theta = \frac 85
.$ [4]
(ii) Solve $\D 15\cos^2 \theta + 2\sin^2\theta = 7$ for $\D 0 \le \theta\le\pi $ radians. [3]
2 (CIE 2012, s, paper 12, question 3)
Show that $\D \cot A + \frac{\sin A}
{1 + \cos A} = \cosec A .$ [4]
3 (CIE 2012, s, paper 22, question 9)
(a) Solve the equation
(i) $\D 3 \sin x – 5 \cos x = 0$ for $\D 0^{\circ } < x < 360^{\circ},$ [3]
(ii) $\D 5 \sin^2 y + 9 \cos y - 3 = 0$ for $\D 0^{\circ} < y < 360^{\circ}.$ [5]
(b) Solve $\D \sin(3 – z) = 0.8$ for $0 < z < \pi$ radians. [4]
4 (CIE 2012, w, paper 11, question 11)
(a) Solve $\D \cosec\left(2x-\frac{\pi}{3}\right)=\sqrt{2}$ for $\D 0<x<\pi$ radians.[4]
(b)(i) Given that $\D 5(\cos y+\sin y)(2\cos y-\sin y)=7,$ show that $\D 12\tan^2y-5\tan y-3=0$[4]
(ii) Hence solve $\D 5(\cos y+\sin y)(2\cos y-\sin y)=7,$ for $\D 0^\circ<x,180^\circ.$[3]
5 (CIE 2012, w, paper 12, question 3)
(i) Show that $\D \cot\theta +\frac{\sin\theta}{1+\cos\theta}=\cosec\theta$.[5]
(ii) Explain why the equation $\D \cot\theta +\frac{\sin\theta}{1+\cos\theta}=\frac{1}{2}$ has no solution.[1]
6 (CIE 2012, w, paper 13, question 10)
(i) Solve $\D \tan^2 x-2\sec x+1=0$ for $\D 0\le x\le 360^\circ.$[4]
(ii) Solve $\D \cos^2 3y=5\sin^2 3y$ for $\D 0\le y\le 2$ radians.[4]
(iii) Solve $\D 2\cosec \left(z+\frac{\pi}{2}\right)=5$ for $\D 0\le z\le 6$ radians.[4]
7 (CIE 2012, w, paper 22, question 11)
(a) Solve $\D 4\sin x+9\cos x=0$ for $\D 0\le x\le 360^\circ$. [3]
(b) Solve $\D \cosec y-1=12\sin y$ for $\D 0<y<360^\circ$.[5]
(c) Solve $\D 3\sec\left(\frac{z}{3}\right)=5$ for $\D 0<z<6\pi $ radians.[4]
8 (CIE 2012, w, paper 23, question 6)
(a) Given that $\D \cos x=p,$ find an expression, in terms of $\D p$, for $\D \tan^2x.$
[3]
(b) Prove that $\D (\cot\theta +\tan\theta)^2=\sec^2\theta+\cosec^2\theta.$
[3]
9 (CIE 2013, s, paper 11, question 11)
(a) Solve $\D 2\sin\left(x+\frac{\pi}{3}\right)=-1$ for $\D 0\le x\le 2\pi.$
[4]
(b) Solve $\D \tan y-2=\cot y$ for $\D 0\le y\le 180^\circ.$
[6]
10 (CIE 2013, s, paper 12, question 3)
Show that $\D (1-\cos\theta-\sin\theta)^2$ $\D -2(1-\sin\theta)(1-\cos\theta)=0.$[3]
11 (CIE 2013, s, paper 12, question 11)
(a) Solve $\D \cos 2x+2\sec 2x+3=0$ for $\D 0\le x\le 360^\circ.$
[5]
(b) Solve $\D 2\sin^2\left(y-\frac{\pi}{6}\right)=1$ for $\D 0\le y\le\pi.$
[4]
12 (CIE 2013, s, paper 21, question 1)
Prove that $\D \left(\frac{1+\sin\theta}{\cos\theta}\right)^2 +
\left(\frac{1-\sin\theta}{\cos\theta}\right)^2 =2+4\tan^2\theta.$
[4]
13 (CIE 2013, w, paper 11, question 3)
Show that $\D \frac{1+\sin\theta}{\cos\theta} +
\frac{\cos\theta}{1+\sin\theta} =2\sec\theta.$
[4]
14 (CIE 2013, w, paper 13, question 3)
Show that $\D \tan^2\theta-\sin^2\theta=\sin^4\theta\sec^2\theta.$[4]
15 (CIE 2013, w, paper 13, question 9)
(a)(i) Solve $\D 6\sin^2x=5+\cos x$ for $\D 0<x<180^\circ.$[4]
(ii) Hence, or otherwise, solve $\D 6\cos^2y=5+\sin y$ for $\D 0<y<180^\circ.$[3]
(b) Solve $\D 4\cot^2z-3\cot z=0$ for $\D 0<z<\pi$ radians.[4]
16 (CIE 2013, w, paper 21, question 12)
(a) Solve the equation $\D 2\cosec x+\frac{7}{\cos x=0}$ for $\D 0\le x\le 360^\circ.$[4]
(b) Solve the equation $\D 7\sin(2y-1=5)$ for $\D 0\le y\le5$ radians.[5]
17 (CIE 2014, s, paper 11, question 1)
Show that $\D \tan\theta+\frac{\cos\theta}{1+\sin\theta}=\sec\theta.$[4]
18 (CIE 2014, s, paper 11, question 11)
(a) Solve $\D 5\sin2x+3\cos2x =0$ for $\D 0\le x\le 180^\circ.$[4]
(b) Solve $\D 2\cot^2y+3\cosec y=0$ for $\D 0\le y\le 360^\circ.$[4]
(c) Solve $\D 3\cos(z+1.2)=2$ for $\D 0\le z\le 6 $ radians.[4]
19 (CIE 2014, s, paper 12, question 1)
Show that $\D \frac{\cos A}{1+\sin a}+\frac{1+\sin A}{\cos A}$ can be written in the form $\D p\sec A$, where $\D p$ is an integer to be found.
20 (CIE 2014, s, paper 12, question 11)
(a) Solve $\D \tan^2x+5\tan x =0$ for $\D 0\le x\le 180^\circ.$[4]
(b) Solve $\D 2\cos^2y-\sin y-1=0$ for $\D 0\le y\le 360^\circ.$[4]
(c) Solve $\D \sec(2z-\frac{\pi}{6})$ for $\D 0\le z\le \pi $ radians.[4]
21 (CIE 2014,s,paper 13, question 9)
Solve
(i) $3\sin x\cos x=2\cos x$ for $0\le x\le 180^{\circ}.$ [4]
(ii) $10\sin^2y+\cos y=8$ for $0\le y\le 360^{\circ}.$ [5]
22 (CIE 2014, s, paper 23, question 7)
(a) Prove that $\dfrac{\tan \theta+\cot \theta}{\sec \theta+\operatorname{cosec} \theta}=\dfrac{1}{\sin \theta+\cos \theta}$. [3]
(b) Given that $\tan x=-\dfrac{5}{12}$ and $90^{\circ}<x<180^{\circ}$, find the exact value of $\sin x$ and of $\cos x$, giving each answer as a fraction.
23 (CIE 2014, w, paper 11, question 11)
(a) Solve $2 \cos 3 x=\cot 3 x$ for $0^{\circ} \leqslant x \leqslant 90^{\circ}$. [5]
(b) Solve $\sec \left(y+\dfrac{\pi}{2}\right)=-2$ for $0 \leqslant y \leqslant \pi$ radians. [4]
$24(\mathrm{CIE} 2014, \mathrm{w}$, paper 13 , question 4)
(a) Solve $3 \sin x+5 \cos x=0$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. [3]
(b) Solve $\operatorname{cosec}\left(3 y+\dfrac{\pi}{4}\right)=2$ for $0 \leqslant y \leqslant \pi$ radians. [5]
25 (CIE 2014, w, paper 21, question 10)
(i) Prove that $\quad \sec x \operatorname{cosec} x-\cot x=\tan x$. $[4]$
(ii) Use the result from part (i) to solve the equation $\sec x \operatorname{cosec} x=3 \cot x$ for $0^{\circ}<x<360^{\circ}$. [4]
26 (CIE 2014, w, paper 23, question 10)
(i) Prove that $\dfrac{1}{1-\cos x}+\dfrac{1}{1+\cos x}=2 \operatorname{cosec}^{2} x$. [3]
(ii) Hence solve the equation $\dfrac{1}{1-\cos x}+\dfrac{1}{1+\cos x}=8$ for $0^{\circ}<x<360^{\circ}$. $[4]$
27 (CIE 2015, s, paper 11, question 10)
(a) Solve $4 \sin x=\operatorname{cosec} x \quad$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. [3]
(b) Solve $\tan ^{2} 3 y-2 \sec 3 y-2=0 \quad$ for $0^{\circ} \leqslant y \leqslant 180^{\circ}$. [6]
(c) Solve $\tan \left(z-\dfrac{\pi}{3}\right)=\sqrt{3} \quad$ for $0 \leqslant z \leqslant 2 \pi$ radians. $[3]$
28 (CIE 2015, s, paper 12, question 2)
Show that $\dfrac{\tan \theta+\cot \theta}{\operatorname{cosec} \theta}=\sec \theta$
29 (CIE 2015, s, paper 12, question 10)
(a) Solve $2 \cos 3 x=\sec 3 x$ for $0^{\circ} \leqslant x \leqslant 120^{\circ}$. [3]
(b) Solve $\quad 3 \operatorname{cosec}^{2} y+5 \cot y-5=0$ for $0^{\circ} \leqslant y \leqslant 360^{\circ}$ [5]
(c) Solve $2 \sin \left(z+\frac{\pi}{3}\right)=1$ for $0 \leqslant z \leqslant 2 \pi$ radians. $[4]$
30 (CIE 2015, w, paper 11, question 3)
Show that $\sqrt{\sec ^{2} \theta-1}+\sqrt{\operatorname{cosec}^{2} \theta-1}=\sec \theta \operatorname{cosec} \theta$.
31 (CIE 2015, w, paper 13, question 2) Solve $2 \cos ^{2}\left(3 x-\dfrac{\pi}{4}\right)=1 \quad$ for $0 \leqslant x \leqslant \dfrac{\pi}{3}$.
32 (CIE 2015, w, paper 21, question 9)
Solve the following equations.
(i) $4 \sin 2 x+5 \cos 2 x=0 \quad$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$ [3]
(ii) $\cot ^{2} y+3 \operatorname{cosec} y=3 \quad$ for $0^{\circ} \leqslant y \leqslant 360^{\circ}$ $[5]$
(iii) $\cos \left(z+\dfrac{\pi}{4}\right)=-\dfrac{1}{2} \quad$ for $0 \leqslant z \leqslant 2 \pi$ radians, giving each answer as a multiple of $\pi$ $[4]$
33 (CIE 2015, w, paper 23, question 8)
(i) Prove that $\sec ^{2} x+\operatorname{cosec}^{2} x=\sec ^{2} x \operatorname{cosec}^{2} x$. [4]
(ii) Hence, or otherwise, solve $\sec ^{2} x+\operatorname{cosec}^{2} x=4 \tan ^{2} x \quad$ for $90^{\circ}<x<270^{\circ}$. $[4]$
34 (CIE 2016, march, paper 12, question 11)
(i) Show that $\dfrac{1}{\operatorname{cosec} \theta-1}-\dfrac{1}{\operatorname{cosec} \theta+1}=2 \tan ^{2} \theta$. [4]
(ii) Hence solve $\dfrac{1}{\operatorname{cosec} \theta-1}-\dfrac{1}{\operatorname{cosec} \theta+1}=6+\tan 0 \quad$ for $0^{\circ}<0<360^{\circ}$. $[4]$
35 (CIE 2016, march, paper 22, question 9)
$P Q R S$ is a quadrilateral with $P S$ parallel to $Q R$. The perimeter of $P Q R S$ is $3 \mathrm{~m}$. The length of $P Q$ is $1 \mathrm{~m}$ and the length of $P S$ is $x \mathrm{~m}$. The point $T$ is on $Q R$ such that $S T$ is parallel to $P Q$. Angle $S R T$ is $\theta$ radians.
(i) Find an expression for $x$ in terms of 0 . [3]
(ii) Show that the area, $A \mathrm{~m}^{2}$, of $P Q R S$ is given by $A=1-\dfrac{\operatorname{cosec} \theta}{2}$.
(iii) Hence find the exact value of $\theta$ when $A=\left(1-\begin{array}{c}\sqrt{3} \\ 3\end{array}\right) \mathrm{m}^{2}$.
36 (CIE 2016, s, paper 11, question 9)
(i) Show that $2 \cos x \cot x+1=\cot x+2 \cos x$ can be written in the form
$(a \cos x-b)(\cos x-\sin x)=0$, where $a$ and $b$ are constants to be found.
(ii) Hence, or otherwise, solve $2 \cos x \cot x+1=\cot x+2 \cos x$ for $0<x<\pi$. $|3|$
37 (CIE 2016, s, paper 12, question 5)
(i) Show that $(1-\cos \theta)(1+\sec \theta)=\sin \theta \tan \theta$
(ii) Hence solve the equation $(1-\cos \theta)(1+\sec \theta)=\sin \theta \quad$ for $0 \leqslant \theta \leqslant \pi$ radians. $[3]$
38 (CIE 2016, s, paper 22, question 12)
Solve the equation
(i) $8 \sin ^{2} A+2 \cos A=7$ for $0^{\circ} \leqslant A \leqslant 180^{\circ}$, [4]
(ii) $\operatorname{cosec}(3 B+1)=2.5$ for $0 \leqslant B \leqslant \pi$ radians. [4]
39 (CIE $2016, \mathrm{w}$, paper 13 , question 8 )
(a) (i) Show that $\dfrac{\operatorname{cosec} \theta}{\operatorname{cosec} \theta-\sin \theta}=\sec ^{2} \theta$. [3]
(ii) Hence solve $\dfrac{\operatorname{cosec} \theta}{\operatorname{cosec} \theta-\sin \theta}=4$ for $0^{\circ}<0<360^{\circ}$. [3]
(b) Sulve $\sqrt{3} \tan \left(x+\dfrac{\pi}{4}\right)=1$ for $0<x<2 \pi$, giving your answers in terms of $\pi$. [3]
40 (CIE 2016, w, paper 21, question 6)
(i) Prove that $\dfrac{\cos x}{1+\tan x}-\dfrac{\sin x}{1+\cot x}=\cos x-\sin x$. $[4]$
(ii) Hence solve the equation $\frac{\cos x}{1+\tan x}-\dfrac{\sin x}{1+\cot x}=3 \sin x-4 \cos x$ for $-180^{\circ}<x<180^{\circ}$. [4]
41 (CIE $2016, \mathrm{w}$, paper 23 , question 6 ) A curve has equation $y=7+\tan x .$ Find
(i) the equation of the tangent to the curve at the point where $x=\frac{\pi}{4}$, $|4|$
(ii) the values of $x$ between 0 and $\pi$ radians for which $\dfrac{\mathrm{d} y}{\mathrm{~d} x}=y$. [4]
42 (CIE 2017, march, paper 12, question 5)
(i) Show that $\operatorname{cosec} \theta-\sin \theta=\cot \theta \cos \theta$. [3]
(ii) Hence solve the equation $\operatorname{cosec} \theta-\sin \theta=\dfrac{1}{3} \cos \theta$, for $0 \leqslant \theta \leqslant 2 \pi$ radians. [4]
43 (CIE 2017, march, paper 22, question 10)
Solve, for $0^{\circ} \leqslant x \leqslant 360^{\circ}$, the equation
(i) $\cot \left(2 x-10^{\circ}\right)=\dfrac{3}{4}$, [4]
(ii) $\sin ^{2} x-\cos ^{2} x=\cos x$. $[5]$
44 (CIE 2017, s, paper 13, question 2)
Given that $y=3+4 \cos 9 x$, write down
(i) the amplitude of $y$, $[1]$
(ii) the period of $y$. [1]
45 (CIE 2017, s, paper 13, question 7)
(a) Show that $\dfrac{\tan ^{2} \theta+\sin ^{2} \theta}{\cos \theta+\sec \theta}=\tan \theta \sin \theta$. [4]
(b) Given that $x=3 \sin \phi$ and $y=\dfrac{3}{\cos \phi}$, find the numerical value of $9 y^{2}-x^{2} y^{2}$.[3]
46 (CIE $2017, \mathrm{~s}$, paper 21 , question 11)
(i) Prove that $\sin x(\cot x+\tan x)=\sec x$[4]
(ii) Hence solve the equation $|\sin x(\cot x+\tan x)|=2$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. [4]
47 (CIE 2017, s, paper 22, question 10) Solve the equation
(i) $4 \sin \left(3 x-\dfrac{\pi}{4}\right)=3$ for $0 \leqslant x \leqslant \dfrac{\pi}{2}$ radians, $[4]$
(ii) $2 \tan ^{2} y+\sec ^{2} y=14 \sec y+3$ for $0^{\circ} \leqslant y \leqslant 360^{\circ}$. $[5]$
48 (CIE 2017, s, paper 23, question 10) Solve the equation
(a) $2|\sin x|=1$ for $-\pi \leqslant x \leqslant \pi$ radians, [3]
(b) $3 \tan \left(2 y+15^{\circ}\right)=1$ for $0^{\circ} \leqslant y \leqslant 180^{\circ}$, [4]
(c) $3 \cot ^{2} z=\operatorname{cosec}^{2} z-7 \operatorname{cosec} z+1$ for $0^{\circ} \leqslant z \leqslant 360^{\circ}$. [5]
49 (CIE 2017, w, paper 11, question 10)
(a) Solve $3 \operatorname{cosec} 2 x-4 \sin 2 x=0$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$. $[4]$
(b) Solve $3 \tan \left(y-\dfrac{\pi}{4}\right)=\sqrt{3}$ for $0 \leqslant y \leqslant 2 \pi$ radians, giving your answers in terms of $\pi$. [4]
50 (CIE $2017, \mathrm{w}$, paper 12 , question 11)
(a) Solve $2 \cot \left(\phi+35^{\circ}\right)=5$ for $0^{\circ} \leqslant \phi \leqslant 360^{\circ}$. [4]
(b) (i) Show that $\dfrac{\sec \theta}{\cot \theta+\tan \theta}=\sin \theta$. [3]
(ii) Hence solve $\dfrac{\sec 3 \theta}{\cot 3 \theta+\tan 3 \theta}=-\dfrac{\sqrt{3}}{2}$ for $-\dfrac{\pi}{2} \leqslant \theta \leqslant \frac{\pi}{2}$, giving your answers in terms of $\pi$. [4]
51 (CIE $2017, \mathrm{w}$, paper 13 , question 1)
Given that $y=2 \sec ^{2} \theta$ and $x=\tan \theta-5$, express $y$ in terms of $x .$
52 (CIE 2017, w, paper 13, question 4) The graph of $y=a \cos (b x)+c$ has an amplitude of 3, a period of $\dfrac{\pi}{4}$ and passes through the point $\left(\dfrac{\pi}{12}, \dfrac{5}{2}\right)$. Find the value of each of the constants $a, b$ and $c$.
53 (CIE 2017, w, paper 21, question 2) Show that $\dfrac{1}{1-\sin \theta}-\dfrac{1}{1+\sin \theta}=2 \tan \theta \sec \theta .$
(a) Show that $\dfrac{\sin x}{1+\cos x}+\dfrac{1+\cos x}{\sin x}=2 \operatorname{cosec} x$.
$[3]$
(b) Solve the following equations.
(i) $\cot ^{2} y+\operatorname{cosec} y-5=0 \quad$ for $0^{\circ} \leqslant y \leqslant 360^{\circ}$
[5]
(ii) $\cos \left(2 z+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{3}}{2}$ for $0 \leqslant z \leqslant \pi$ radians
55(CIE 2018, march, paper 12, question 3)
Do not use a calculator in this question.
(a) Simplify $\dfrac{(3+2 \sqrt{5})(6-2 \sqrt{5})}{(4-\sqrt{5})}$, giving your answer in the form $a+b \sqrt{5}$, where $a$ and $b$ are $\begin{array}{ll}\text { integers. } & {[3]}\end{array}$
(b) In this part, all lengths are in centimetres.
The diagram shows the triangle $A B C$ with $A B=6-2 \sqrt{3}$ and $B C=6+2 \sqrt{3}$. Given that
$\cos A B C=-\dfrac{1}{2}$, find the length of $A C$ in the form $c \sqrt{d}$, where $c$ and $d$ are integers.
56 (CIE 2018, march, paper 22, question 11)
(a) (i) Show that $\dfrac{(1-\sin A)(1+\sin A)}{\sin A \cos A}=\cot A$.
[2]
(ii) Hence solve $\dfrac{(1-\sin 3 x)(1+\sin 3 x)}{\sin 3 x \cos 3 x}=\dfrac{1}{2}$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$.
[4]
(b) Solve $10 \tan ^{2} y-\sec y-1=0$ for $0 \leqslant y \leqslant 2 \pi$ radians.
[5]
57 (CIE 2018, s, paper 12, question 8)
(a) Solve $3 \cos ^{2} \theta+4 \sin \theta=4$ for $0^{\circ} \leqslant \theta \leqslant 180^{\circ}$.
[4]
(b) Solve $\sin 2 \phi=\sqrt{3} \cos 2 \phi$ for $-\dfrac{\pi}{2} \leqslant \phi \leqslant \dfrac{\pi}{2}$ radians.
$[4]$
$58(\mathrm{CIE} 2018, \mathrm{~s}$, paper 12, question 10) Do not use a calculator in this question.
All lengths in this question are in centimetres.
The diagram shows the triangle $A B C$, where $A B=4 \sqrt{3}-5, B C=4 \sqrt{3}+5$ and angle $A B C=60^{\circ}$. It is known that $\sin 60^{\circ}=\dfrac{\sqrt{3}}{2}, \cos 60^{\circ}=\dfrac{1}{2}, \tan 60^{\circ}=\sqrt{3}$.
(i) Find the exact value of $A C$.
(ii) Hence show that $\operatorname{cosec} A C B=\dfrac{2 \sqrt{p}}{q}(4 \sqrt{3}+5)$, where $p$ and $q$ are integers.
59 (CIE 2018, s, paper 21, question 11)
(a) Solve $10 \cos ^{2} x+3 \sin x=9$ for $0^{\circ}<x<360^{\circ}$.$[5]$
(b) Solve $3 \tan 2 y=4 \sin 2 y$ for $0<y<\pi$ radians.$[5]$
60 (CIE 2018, s, paper 22 , question 1)
(i) Show that $\cos \theta \cot \theta+\sin \theta=\operatorname{cosec} \theta$.[3\rfloor
(ii) Hence solve $\cos \theta \cot \theta+\sin \theta=4$ for $0^{\circ} \leqslant \theta \leqslant 90^{\circ}$.[2]
Answers
Answers
$1. (ii)0.902; 2.24\\
2. \mbox{Proof}\\
3. (ai) 59; 239\\
(ii)101.5,258.5\\
(b)0.786\\
4. (a)x = 7\pi/24; 13\pi/24\\
(bii)36.9; 161.6\\
5. (ii)\sin \theta= 2 \mbox{No solution}\\
6. (i)x = 60; 300\\
(ii)y = .14; .907; 1.19; 1.95\\
(iii)z = 1.94; 5.91\\
7. (a)114,294\\
(b)14.5,199.5,165.5340.5\\
(c)2.78,16.1\\
8. (a)(1 - p)/p^2\\
9. (a)x = 5\pi/6; 3\pi/2\\
(b)y = 67.5; 157.5\\
10. \mbox{Proof}\\
11. (a)90; 270\\
(b)5\pi/12; 11\pi/12\\
12. \mbox{Proof}\\
13. \mbox{Proof}\\
14. \mbox{Proof}\\
15. (ai)x = 70.5; 120\\
(ii)y = 19.5; 160.5\\
(iii)z = 0.927\\
16. (a)164.1,344.1\\
(ii)0.898,1.67,4.04,4.81\\
17. \mbox{Proof}\\
18. (a)x = 74.5; 164.5\\
(b)y = 210; 330\\
(c)z = 4.24; 5.92\\
19. p = 2\\
20. (a)x = 0; 180; 101.3\\
(b)y = 270\\
(c)z = 11\pi/12\\
$
21. (i) $x=41.8,138.2$
(ii) $y=60,300,113.6,246.4$
22. (b)sin $x=5 / 13, \cos x=-12 / 13$
23. (a) $30,90,10,50$
(b) $y=\pi / 6,5 \pi / 6$
24. (a) $x=121.0,301.0$
(b) $y=7 \pi / 36,23 \pi / 36,31 \pi / 36$
25. (ii) $54.7,125.3,234.7,305.3$
26. (ii) $x=30,150,210,330$
27. (a) $30,150,210,330$
(b) $60,180,23.5,96.5,143.5$
(c) $2 \pi / 3,5 \pi / 3$
28. proved
29. (a) $15,45,75,105$
(b) $71.6,251.6,253.4,333.4$
(c) $\pi / 2,11 \pi / 6$
30. proof
31. $0, \pi / 6, \pi / 3$
32. (i) $x=64.3,154.3$
(ii) $y=194.5,345.5,90$
(iii) $z=5 \pi / 12,13 \pi / 12$
33. (ii) $x=135,225$
34. (ii) $63.4,123.7,243.4,303.7$
35. (i) $x=1-1 /(2 \tan \theta)-1 / 2 \sin \theta$
(iii) $\theta=\pi / 3$
36. (i) $a=2, b=1$
(ii) $\pi / 3, \pi / 4$
37. (ii) $\pi / 4,0, \pi$
38. (i) $60,104.477$
(ii) $0.577,1.90,2.67$
39. (aii) $60,120,240,300$
(b) $-\frac{\pi}{12}, \frac{11 \pi}{12}, \frac{23 \pi}{12}$
40. (ii) $51.3,-128.7$
41. (i) $y=2 x+6.429$
(ii) $x=1.25,2.03$
42. proved
43. $(\mathrm{i}) 31.6,121.6,211.6,301.6$
(ii) $60,300,180$
44. 4,40 or $2 \pi / 9$
45. (b)81
46. (ii) 120,240
47. (i) $1.03,0.544$
(ii) $78.5,281.5$
48. (a) $\pm \pi / 6, \pm 5 \pi / 6$
(b) $1.7,91.7$
(c) $194.5,345.5$
49. (a) $30,60,120,150$
(b) $5 \pi / 12,17 \pi / 12$
50. (a) $166.8,346.8$
(bii) $-2 \pi / 9,-\pi / 9,4 \pi / 9$
51. $y=2(x+5)^{2}+2$
52. $a=3, b=8, c=4$
53. proved
54. (bi) $y=30,150,199.5,340.5$
(bii) $7 \pi / 24,11 \pi / 24$
55. (a) $2 \sqrt{5}+2$
(b) $2 \sqrt{30}$
56. $21.1,81.1,141.1$
$\pi, 0.430,5.85$
57. (a) $19.5,160.5,90$
(b) $\frac{\pi}{6}$
58. (i) $\sqrt{123}$
(ii) $\frac{2 \sqrt{41}}{23}(4 \sqrt{3}+5)$
59. (a) $30,150,191.5,348.5$
(b) $\pi / 2, .361,2.78$
60. (ii) $14.5$
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