Monday, December 3, 2018

Trigonometry

$\def\D{\displaystyle}\def\cosec{\text{cosec }}$
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]

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\\
$





No comments:

Post a Comment