$\def\frac{\dfrac}$
Group (2015-2019)
1.(2015/Myanmar,q5)
If $\alpha+\beta+\gamma=
180^{\circ}$, prove that $\sin \frac{\alpha+\beta}{2}=\sin \left(90^{\circ}+\frac{\gamma}{2}\right)$. (3 marks)
2.(2015/Myanmar,q14a)
Without the use of table evaluate $\tan (\alpha+\beta+\gamma)$, given that $\tan \alpha=\frac{1}{2}$; $\tan \beta=\frac{1}{3}$ and $\tan \gamma=\frac{1}{4}$. $\quad$ (5 marks)
3.(2015/Myanmar,q14b)
$A, B, C$ are three towns, $B$ is 10 miles from $A$ in a direction $N 47^{\circ} E . C$ is 17 miles away from $B$ in a direction $N 70^{\circ} W$. Calculate the distance and direction of $A$ from $C . \quad(5$ marks $)$
4.(2015/FC,q3)
Given that $\sin ^{2} x, \cos ^{2} x$ and $5 \cos ^{2} x-3 \sin ^{2} x$ are in A.P., find the value of $\sin ^{2} x . \quad(3$ marks $)$
5.(2015/FC,q5)
Prove the identity $\cos 3 \theta-\cos \theta=-4 \sin ^{2} \theta \cos \theta .$ (3 marks)
6.(2015/FC,q14a) Show that $\sin (\alpha+\beta) \cdot \sin (\alpha-\beta)=\sin ^{2} \alpha-\sin ^{2} \beta.\quad$ (5 marks)
7.(2015/FC,q14b)
Given that $\sin A=\frac{2}{\sqrt{5}}, \cos B=-\frac{\sqrt{2}}{3}$ and that both $A$ and $B$ are in the same quadrant, calculate tbe value of each of the following: $\begin{array}{lll}\text { (i) } \cos (A+B) & \text { (ii) } \cos (2 A-B) & (5 \text { marks })\end{array}$
8.(2016/Myanmar,q3)
$\def\omit{The ninth term of an arithmetic progression is 6 . Find the sum of the first 17 terms.}$
9.(2016/Myanmar,q5)
Given that $A=B+C$, prove that $\tan A-\tan B-\tan C=\tan A \tan B \tan C$.
10.(2016/Myanmar,q14a)
If $\cot x+\cos x=p$ and $\cot x-\cos x=q$, show that $\sqrt{p q}=\cos x \cot x$, where $x$ is acute and hence, prove that $p^{2}-q^{2}=4 \sqrt{p q}$
11.(2016/Myanmar,q14b)
A man travels $10 \mathrm{~km}$ in a direction $\mathrm{N} 70^{\circ} \mathrm{E}$ and then $5 \mathrm{~km}$ in a direction N $40^{\circ} \mathrm{E}$. What is his final distance and bearing from his starting point?
12.(2016/FC,q5)
If $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$, prove that $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$.
13.(2016/FC,q14a)
Given that $\frac{\cos (\alpha-\beta)}{\cos (\alpha+\beta)}=\frac{7}{5}$, prove that $\cos \alpha \cos \beta=6 \sin \alpha \sin \beta$ and deduce a relationship between $\tan \alpha$ and $\tan \beta .$ Given further that $\alpha+\beta=45^{\circ}$, calculate the value of $\tan \alpha+\tan \beta$.
14.(2016/FC,q14b)
A town $P$ is 25 miles away from the town $Q$ in the direction $N 35^{\circ} \mathrm{E}$ and a town $R$ is 10 miles from $Q$ in the direction $42^{\circ}$ W. Calculate the distance and bearing of $P$ from $R$.
15.(2017/Myanmar,q5)
Prove that $\sin x+\sin 2 x+\sin 3 x=\sin 2 x(1+2 \cos x)$.
Q(5) Solution
16.(2017/Myanmar,q14a)
Prove the identity $\sec 2 \alpha=\frac{1+\tan ^{2} \alpha}{2-\sec ^{2} \alpha}$.
Q14(a) Solution
17.(2017/Myanmar,q14b)
A town $P$ is $50 \mathrm{~km}$ away from a town $Q$ in the direction $N 35^{\circ} E$ and a town $R$ is $68 \mathrm{~km}$ from $Q$ in the direction $N 42^{\circ} 12^{\prime} W$. Calculate the distance and bearing of $P$ from $R$.
Q14(b) Solution
18.(2017/FC,q5)
Prove that $\frac{1+\cos x+\cos 2 x}{\sin x+\sin 2 x}=\cot x$. (3 marks)
19.(2017/FC,q14a)
Solve the equation $ \sin x+\sin \frac{x}{2}=0 \text { for } 0 \leq x \leq 2 \pi.$ (5 marks)
20.(2017/FC,q14b)
$\mathrm{A}$ and $\mathrm{B}$ are two points on one bank of a straight river, distant from one another $649 \mathrm{~m} . \mathrm{C}$ is on the other bank and the measures of the angles $\mathrm{CAB}, \mathrm{CBA}$ are respectively $48^{\circ} 31^{\prime}$ and $75^{\circ} 25^{\prime} .$ Find the width of the river. $\quad(5$ marks $)$
21.(2018/Myanmar,q5)
If $A+B=45^{\circ}$, show that $\tan A+\tan B+\tan A \tan B=1$.
Click for Solution
22.(2018/Myanmar,q14a)
Two acute angles, $\alpha$ and $\beta$, are such that $\tan \alpha=\frac{4}{3}$ and $\tan (\alpha+\beta)=-1$. Without evaluating $\alpha$ or $\beta$, show that $\tan \beta=7$, evaluate $\sin \alpha$ and $\sin \beta$.
Click for Solution
23.(2018/Myanmar,q14b)
A ship is $5 \mathrm{~km}$ away from a boat in a direction $N 37^{\circ} \mathrm{W}$ and a lighthouse is $12 \mathrm{~km}$ away from the boat in a direction $S 53^{\circ} W .$ Calcualate the distance and direction of the ship from the lighthouse.
Click for Solution
24.(2018/FC,q5)
Prove that $\frac{1-\cos 2 x+\sin 2 x}{1+\cos 2 x+\sin 2 x}=\tan x$.
25.(2018/FC,q14a)
If $\alpha+\beta+\gamma=180^{\circ}$, show that $$\cos \frac{\alpha}{2}+\cos \frac{\beta}{2}+\cos \frac{\gamma}{2}=4 \cos \frac{\beta+\gamma}{4} \cos \frac{\gamma+\alpha}{4} \cos \frac{\alpha+\beta}{4}$$
26.(2018/FC,q14b)
In $\triangle A B C, c=10, b=6$ and $a=5$. Check whether $\angle A C B$ is acute or obtuse and find its magnitude.
27.(2019/Myanmar,q5a)
Solve the equation $2 \cos x \sin x=\sin x$ for $0^{\circ} \leq x \leq 360^{\circ}$. (3. marks) Click for Solution
28.(2019/Myanmar,q12b)
If $\alpha+\beta+\gamma=180^{\circ}$, prove that $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\tan \frac{\beta}{2} \tan \frac{\gamma}{2}+\tan \frac{\alpha}{2} \tan \frac{\gamma}{2}=1$. (5marks) Click for Solution
29.(2019/Myanmar,q13a)
In $\triangle \mathrm{ABC}$, if $\angle \mathrm{B}=\angle \mathrm{A}+15^{\circ}, \angle \mathrm{C}=\angle \mathrm{B}+15^{\circ}$ and $\mathrm{BC}=6$, find $\mathrm{AC}$. $\quad$ (5 marks) Click for Solution
30.(2019/FC,q5a)
$\begin{array}[t]{ll}\text { Solve the equation } \cos ^{2} x=2+\cos x \text { for } 0^{*} \leq x \leq 360^{\circ} . & \text { (3 marks) }\end{array}$ Click for Solution 5(a)
31.(2019/FC,q12b)
If $\alpha+\beta+\gamma=180^{\circ}$, prove that $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=4 \sin \alpha \sin \beta \sin \gamma$. (5 marks)Click for Solution 12(b)
32.(2019/FC,q13a)
In $\Delta A B C$, if $\angle A: \angle B: \angle C=1: 3: 8$ and $A B=9$, find $A C$Click for Solution 13(a)
Answer (2015-2019)
1. Prove
2. (a) $5/3$
3. (b) 15.32, S $34^{\circ}26'$ E
4. $\frac 35$
5. Prove
6. (a) Prove
7. (b) $\cos(A+B)=\frac{\sqrt{10}-2\sqrt{35}}{15}$ $\cos(2A-B)=\frac{3\sqrt 2-4\sqrt 7}{15}$
8. $S_{17}=102
9. Prove
10. (a) Prove
11. (b) $14.55 \mathrm{~km}, \mathrm{~N} 60^{\circ}{6}^{\prime} \mathrm{E}$
12. Prove
13. (a) $\frac{5}{6} \quad$
14. (b) $612,5, N 58^{\circ} 15^{\prime} $ E
15. Prove
16. (a) Prove
17. (b) $74.95 \mathrm{~km}, 58247^{\prime} E$
18. Prove
19. (a) $0,2\pi,\dfrac{4\pi}{3}$
20. (b) 567
21. Show
22. (a) $\sin \alpha=\frac{4}{5}, \sin \beta=\frac{7 \sqrt{2}}{10}$
23. (b) $13 \mathrm{~km}$, $N 30^{\circ} 23^{\prime} E$
24. Prove
25. (a) Prove
26. (b) Obtuse, $130^{\circ} 33^{\prime}$
27. (a) $x=0^{\circ},60^{\circ}, 180^{\circ}, 300^{\circ}, 360^{\circ}$
28. (b) Proof
29. (a) $3\sqrt 6$
30. (a) $180^{\circ}$
31. (b) Prove
32. (a) $3\sqrt 6$
Group (2014)
1. In $\triangle A B C$, if $\alpha: \beta: \gamma=2: 3: 7$ and $b=8$, find $a$. (3 marks)
2. Solve $\triangle A B C$ with $\alpha=35^{\circ}, \beta=68^{\circ}$ and $c=25$. (5 marks)
3. Solve $\triangle A B C$ if $\angle A=64^{\circ} 20^{\prime}, \angle B=50^{\circ}, b=15$. (5 marks)
4. Solve that $\triangle A B C$ with $a=20, c=18, \beta=45^{\circ}$. (5 marks)
5. Solve the triangle $A B C$ with $b=18.1, c=12.3$ and $\alpha=75^{\circ}$. (5 marks)
6. Solve $\triangle A B C$ with $b=18.1, c=12.3$ and $\alpha=115^{\circ}$. (5 marks)
7. Solve $\triangle A B C$ with $a=13, b=10, c=9$. (5 marks)
8. Solve $\triangle A B C$ if $B C=10, A C=15$ and $\angle A=30^{\circ}$. (5 marks)
9. Given that $\sin \theta=-\frac{5}{13}$ where $180^{\circ}<\theta<270^{\circ}$, find the values of $\tan(\theta)$ and $\cos (\theta).$ (3 marks)
10. Find all the angles between $0^{\circ}$ and $360^{\circ}$ which satisfy the equation, $\sin \left(2 \theta+15^{\circ}\right)=\frac{\sqrt{3}}{2}$. (3 marks)
11. Given that $\tan 2 A=\frac{120}{119}$ and that $\angle A$ is acute, find without using tables, the values of $\cos 2 A$ and $\sin 2 A$. (5 marks)
12. Given that $x=3 \sin \theta-2 \cos \theta$ and $y=3 \cos \theta+2 \sin \theta$, find the value of the acute angle $\theta$ for which $x=y$. Show also that $x^{2}+y^{2}$ is constant for all values of $\theta$. (5 marks)
13. Solve the equation $2 \cos ^{2} x-\sin x-1=0$ for $0^{\circ} \leq x \leq 360^{\circ}$. (3 marks)
14. Solve the equation $\cos ^{2} 2 x-\sin ^{2} 2 x=0$ for $0 \leq x<2 \pi$. (3 marks)
15. Solve the equation $\cos x+\cos 3 x=0$ for $0^{\circ} \leq x \leq 360^{\circ}$. (3 marks)
16. Solve the trigonometric equation $2 \cos 2 x+1=0,-\pi<x \leq \pi$. (5 marks)
17. If $\tan x=a, \tan 2 x=3 a$ and $0^{\circ}<x<90^{\circ}$, then find $x$. (3 marks)
18. A man walking along a straight road $P Q X$ observes two hills $A, B$ on the left of the road. When he is at $P$; he sees the hills at $\angle X P A=25^{\circ}, \angle X P B=40^{\circ} .$ When he reaches $Q, 1$ miles along the road from $P$, he sees the hills in a line at $\angle X Q A=63^{\circ}$. Find $A B$. (5 marks)
19. A ship sails 15 miles on a course of $S 40^{\circ} 10^{\prime} W$ and then 21 miles on a course $N 28^{\circ} 20^{\prime} W$. Find the distance and direction of the last position from the first. (5 marks)
20. Prove that $\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x}=2 \sec x$. (3 marks)
21. Prove the identity $\frac{\cot A-\tan A}{\cot A+\tan A}=\cos ^{2} A-\sin ^{2} A$. (3 marks)
22. Prove the identity $\frac{\sec ^{2} \theta-2}{\sec ^{2} \theta+2 \tan \theta}=\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}$. (3 marks)
23. Show that $\sin \left(\theta+\frac{\pi}{6}\right)-\cos \left(\theta+\frac{\pi}{3}\right)=\sqrt{3} \sin \theta$. (3 marks)
24. If $\theta+\phi+\delta=\pi$, show that $\tan \theta+\tan \phi+\tan \delta=\tan \theta \tan \phi$ tan $\delta$. (5 marks)
25. Prove that $\frac{\sin x}{\sec x-1}+\frac{\sin x}{\sec x+1}=2 \cot x$. (5 marks)
26. Prove that $\frac{\tan \theta+\tan \phi}{\tan \theta-\tan \phi}=\frac{\sin (\theta+\phi)}{\sin (\theta-\phi)}$. Use this result, to calculate the value of the ratio $\sin 75^{\circ}: \sin 15^{\circ}$ in surd form. (5 marks)
27. Prove that $\frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta}=\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)} .$ By using this result show that $\frac{\sin 105^{\circ}}{\sin 15^{\circ}}=2+\sqrt{3}$. (5 marks)
28. Prove that $\tan A+\cot A=2 \operatorname{cosec} 2 A$. Find also, without using tables, the exact value of $\tan 112.5^{\circ}+\cot 112.5^{\circ}$. (5 marks)
29. If $\tan B=3 \tan C$, prove that $\tan (B-C)=\frac{\sin 2 C}{2-\cos 2 C}$. (5 marks)
30. If $\alpha+\beta+\theta=\pi$, show that $\sin \alpha-\sin \beta+\sin \theta=4 \sin \frac{\alpha}{2} \cos \frac{\beta}{2} \sin \frac{\theta}{2}$. (5 marks)
31. Show that $\cos \left(\theta-40^{\circ}\right)=\sin \left(\theta+50^{\circ}\right) .$ Hence or otherwise, solve the equation $\cos \left(\theta-40^{\circ}\right)=4 \sin \left(\theta+50^{\circ}\right)$, giving all solutions between $0^{\circ}$ and $360^{\circ}$. (5 marks)
32. If $A, B, C$ are the angles of a triangle and $\tan A=1$ and $\tan B=2$, prove without using tables or calculators that $\tan C=3$. If $a, b, c$ are the corresponding sides of the triangle, prove that $\frac{a}{\sqrt{5}}=\frac{b}{2 \sqrt{2}}=\frac{c}{3}$. (5 marks)
Answer (2014)
1. $a=4 \sqrt{2}$
2. $r=77^{\circ}, a=14.72, b=23.79$
3. $\angle C=65^{\circ} 40^{\prime}, c=17.84, a=7.707$
4. $b=14.659, r=60^{\circ} 19^{\prime}, \alpha=74^{\circ} 41^{\prime}$
5. $a=19.07, r=38^{\circ} 32^{\prime}, \beta=66^{\circ} 28^{\prime}$
6. $a=25.83, \beta=39^{\circ} 26^{\prime}, r=25^{\circ} 34^{\prime}$
7. $\alpha=96^{\circ} 11^{\prime}, \beta=50^{\circ} 8^{\prime}, \gamma=43^{\circ} 41^{\prime}$
8. $c_{1}=6.318, \gamma_{1}=18^{\circ} 35^{\prime}, \gamma_{2}=131^{\circ} 25^{\prime}$ $c_{2}=19.604, \beta_{2}=101^{\circ} 25^{\prime}, \beta_{1}=48^{\circ} 35^{\prime}$
9. $\frac{5}{12},-\frac{12}{13} \quad $
10. $\theta=22.5^{\circ}$ (or) $52.5^{\circ}$ (or) $202.5^{\circ}$ (or) $232.5^{\circ}$
11. $\cos 2 A=\frac{119}{169}, \sin 2 A=\frac{120}{169}$
12. $\theta=78^{\circ} 42^{\prime}$
13. $x=30^{\circ}$ (or) $150^{\circ}$ (or) $270^{\circ}$
14. $x=\frac{\pi}{8}$ (or) $\frac{3 \pi}{8}$ (or) $\frac{5 \pi}{8}$ (or) $\frac{7 \pi}{8}$ (or) $\frac{9 \pi}{8}($ or $) \frac{11 \pi}{8}($ or $) \frac{13 \pi}{8}$ (or $) \frac{15 \pi}{8}$
15. $x=45^{\circ}$ (or) $90^{\circ}$ (or) $135^{\circ}$ (or) $225^{\circ}$ (or) $315^{\circ}$ (or) $270^{\circ} \quad$
16. $x=\frac{\pi}{3}$ (or) $\frac{2 \pi}{3}$ (or) $-\frac{\pi}{3}$ (or) $-\frac{2 \pi}{3}$
17. $x=30^{\circ}$
18. $1.320$ miles
19. $20.86$ miles, $N 70^{\circ} 22^{\prime} W$
20. Prove
21. Prove
22. Prove
23. Prove
24. Prove
25. Prove
26. $(2+\sqrt{3}): 1$
27. Prove
28. $-2 \sqrt{2}$
29. Prove
30. Prove
31. $\theta=130^{\circ}$ (or $) 310^{\circ}$
32. Prove
Group (2013)
Group (2012)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | In $\triangle A B C, a=\sqrt{7}, b=2, c=1$, find the largest angle.(3 marks) | |
| 2. | In triangle $A B C, a: b: c=1: 3: \sqrt{7}$, find $\angle C$.(3 marks) | |
| 3. | In $\triangle A B C$, if $A B=x, B C=x+2$ and $A C=x-2$ where $x>4$, prove that $\cos A=\frac{x-8}{2(x-2)}$.Find the integral values of $x$ for which $A$ is obtuse.(5 marks) | |
| 4. | Given that $\sin \alpha=-\frac{4}{5}, \cos \beta=-\frac{12}{13}$ and that $\alpha$ and $\beta$ are in the same quadrant, find without using table, the value of $\tan (\alpha-\beta)$.(5 marks) | |
| 5. | Given that $\sin \alpha=-\frac{3}{5}, \tan \beta=\frac{5}{12}$ and that $\alpha$ and $\beta$ are in the same quadrant.Find the values of $\sin (\alpha-\beta)$ and tan $(\alpha+\beta)$ without using tables.(5 marks) | |
| 6. | Given that $\cos A=\frac{12}{13}$ and $A$ is in quadrant $I$, and that $\sin B=\frac{4}{5}$ and $B$ is in quadrant $I I$, find $\tan (A+B)$ and find also the quadrant in which the terminal side of $(A+B)$ lies.(5 marks) | |
| 7. | Given that $\tan x=\frac{5}{12}, \sin y=-\frac{12}{13}$ and $x$ and $y$ are in the same quadrant.Find the value of $\frac{\cos (x-y)}{\sin (x+y)}$.(5 marks) | |
| 8. | $\tan A=-\frac{3}{4}, \cos B=-\frac{5}{13}, A$ and $B$ are between $180^{\circ}$ and $360^{\circ}$.Without using tables, find values of $\sin (A-B), \tan (A-B)$.(5 marks) | |
| 9. | $\tan \alpha=\frac{1}{2}, \tan (\alpha+\beta)=1$ and $\tan (\alpha+\beta+\lambda)=\frac{5}{3}$.Without using tables calculate the values of $\tan \beta$ and $\cot 2 \lambda$.(5 marks) | |
| 10. | Given that $\tan \alpha=p$ and $\tan (\alpha-\beta)=q$, express $\tan \beta$ in terms of $p$ and $q$.Calculate the value of $\tan (\alpha+\beta)$ when $p=1$ and $q=0.5$.(5 marks) | |
| 11. | Prove that $\tan \alpha+\cot \alpha=2 \operatorname{cosec} 2 \alpha$.By using this equation, find the value of $\tan 75^{\circ}+\tan 15^{\circ}$.(5 marks) | |
| 12. | Prove the identity $\frac{\tan ^{2} \alpha+1}{\tan ^{2} \alpha-1}=-\sec (2 \alpha)$.(3 marks) | |
| 13. | Show that $\frac{1+\tan x}{1-\tan x}=\sec 2 x+\tan 2 x$.(3 marks) | |
| 14. | Prove the identity $\frac{\sin x}{1+\cos x}=\tan \frac{x}{2}$.(3 marks) | |
| 15. | Prove the identity $\sin \theta \sec \theta+\cos \theta \operatorname{cosec} \theta=2 \operatorname{cosec} 2 \theta$.(3 marks) | |
| 16. | If $\sin x+\sin ^{2} x=1$, then prove that $\cos ^{8} x+2 \cos ^{6} x+\cos ^{4} x=1$.(3 marks) | |
| 17. | Show that $\sin \frac{\pi}{12} \cos \frac{5 \pi}{12}=\frac{2-\sqrt{3}}{4}$.(5 marks) | |
| 18. | Prove that $\tan 2 x(\operatorname{cosec} x-2 \sin x)=2 \cos x$.(5 marks) | |
| 19. | Prove that $\tan 2 \theta(2 \cos \theta-\sec \theta)=2 \sin \theta$.(5 marks) | |
| 20. | If $\sin x+\cos x=a$, then prove that $\sin ^{6} x+\cos ^{6} x=\frac{1}{4}\left[4-3\left(\mathrm{a}^{2}-1\right)^{2}\right]$.(5 marks) | |
| 21. | In $\triangle A B C, a=x-2, b=x+2$ and $c=x$.Prove that $\cos \alpha=\frac{x+8}{2 x+4}$.If $x=5$, find $\alpha$.(5 marks) | |
| 22. | Solve the equation $\sin ^{2} \theta-\sin \theta=2$ for $0^{\circ} \leq \theta \leq 360^{\circ}$.(3 marks) | |
| 23. | Solve the equation $\cos 2 x=\sin x$ for $0^{\circ} \leq x<360^{\circ}$.(3 marks) | |
| 24. | Solve the equation $2 \sin ^{2} x-\cos x-1=0$ for $0^{\circ} \leq x \leq 360^{\circ}$.(3 marks) | |
| 25. | Solve the equation $\cos (2 x+\pi)=\sin \left(\frac{3 \pi}{2}-x\right)$ for $0 \leq x<2 \pi$.(3 marks) | |
| 26. | Solve the triangle $A B C$, where $a=4, b=7$ and $c=5$.(5 marks) | |
| 27. | Solve the triangle $a=6.1, b=4.1, c=3.1$.(5 marks) | |
| 28. | Solve the $\triangle A B C$ with $\alpha=52^{\circ} 30^{\prime}, b=11$ and $c=7$.(5 marks) | |
| 29. | Solve the $\triangle A B C$ with $\gamma=60^{\circ}, a=5, b=8$.(5 marks) | |
| 30. | Solve $\triangle A B C$ with $\gamma=30^{\circ}, \alpha=135^{\circ}$ and $a=100$.(5 marks) | |
| 31. | Solve $\triangle A B C$ with $a=5, \beta=75^{\circ}$ and $\gamma=41^{\circ}$.(5 marks) | |
| 32. | In $\triangle A B C \angle A=108^{\circ} 12^{\prime}, \angle B=50^{\circ} 18^{\prime}$ and $c=15$.Solve the triangle.(5 marks) | |
| 33. | A ship $A$ is $6 \mathrm{~km}$ from a lighthouse, $C$ in a direction $N 30^{\circ} W$ and a boat $B$ is $11 \mathrm{~km}$ from that lighthouse $C$ in the direction $S 50^{\circ} W$.Calculate the distance and direction of $A$ from $C$.(5 marks) |
Answer (2012)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $120^{\circ}$ | |
| 2. | $60^{\circ}$ | |
| 3. | $5,6,7$ | |
| 4. | $\frac{33}{56}$ | |
| 5. | $\frac{16}{65}, \frac{56}{33}$ | |
| 6. | $-\frac{33}{56}$, quadrant II | |
| 7. | $\frac{120}{169}$ | |
| 8. | $\frac{63}{65}, \frac{63}{16}$ | |
| 9. | $\frac{1}{3}, \frac{15}{8}$ | |
| 10. | $\frac{p-q}{1+p q}, 2$ | |
| 11. | 4 | |
| 12. | Proof | |
| 13. | Proof | |
| 14. | Proof | |
| 15. | Proof | |
| 16. | Proof | |
| 17. | Proof | |
| 18. | Proof | |
| 19. | Proof | |
| 20. | Proof | |
| 21. | $21^{\circ} 47^{\prime}$ | |
| 22. | $270^{\circ}$ | |
| 23. | $30^{\circ}, 150^{\circ}, 270^{\circ}$ | |
| 24. | $60^{\circ}, 180^{\circ}, 300^{\circ}$ | |
| 25. | $0, \frac{2 \pi}{3}, \frac{4 \pi}{3}$ | |
| 26. | $\alpha=34^{\circ} 3^{\prime}, \beta=101^{\circ} 31^{\prime}, \gamma=44^{\circ} 26^{\prime}$ | |
| 27. | $\alpha=115^{\circ} 6^{\prime}, \beta=37^{\circ} 30^{\prime}, \gamma=27^{\circ} 24^{\prime}$ | |
| 28. | $a=8.732, \beta=88^{\circ} 12^{\prime}, \gamma=39^{\circ} 18^{\prime}$ | |
| 29. | $c=7, \alpha=38^{\circ} 13^{\prime}, \gamma=81^{\circ} 47^{\prime}$ | |
| 30. | $\beta=15^{\circ}, b=36.6, c=50 \sqrt{2}$ | |
| 31. | $\alpha=64^{\circ}, b=5.373, c=3.650$ | |
| 32. | $\angle C=21^{\circ} 30^{\prime}, a=38.88, b=31.49$ | |
| 33. | $13.41 \mathrm{~km}, N 23^{\circ} 51^{\prime} E$ |
Group (2011)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | Given that $\tan \alpha=\frac{1}{3}, \tan \beta=\frac{1}{4}$ and $\tan \gamma=\frac{1}{6}$, without the use of table evaluate $\tan (\alpha+\beta-\gamma)$. $\mbox{ (3 marks)}$ | |
| 2. | Given that $\sin \theta=-\frac{5}{13}$ where $180^{\circ} < \theta < 270^{\circ}$, find the values of $\tan \theta, \sin \frac{1}{2} \theta$ and $\cos (-\theta)$. $\mbox{ (5 marks)}$ | |
| 3. | If $\sin \theta=\frac{3}{5}, \cos \beta=\frac{5}{13}$ where $\theta$ and $\beta$ are acute angles, find $\sin 2(\theta+\beta)$. $\mbox{ (5 marks)}$ | |
| 4. | Given that $\sin \alpha=\frac{8}{17}$ and $\cos \beta=\frac{4}{5}$ and that $\alpha$ and $\beta$ are in the same quadrant.Without using tables, find the values of $\tan (\alpha+\beta)$ and $\tan (\alpha-\beta)$. $\mbox{ (5 marks)}$ | |
| 5. | Given that $\sin \alpha=\frac{15}{17}, \cos \beta=\frac{-3}{5}$ and that $\alpha$ and $\beta$ are in the same quadrant, find without using tables the values of $\tan 2 \alpha$ and $\tan (2 \alpha+\beta)$. $\mbox{ (5 marks)}$ | |
| 6. | Given that $\sin \alpha=\frac{5}{13}$, where $90^{\circ} \leq \alpha \leq 180^{\circ}$ and that $\cos \beta=-\frac{3}{5}$ where $180^{\circ} \leq \beta \leq 360^{\circ}$, find the values of $\sin 2 \beta$ and $\sin (\alpha+2 \beta)$. $\mbox{ (5 marks)}$ | |
| 7. | Find exact value of $4 \sin \frac{\pi}{24} \cos \frac{\pi}{24} \cos \frac{\pi}{12}$. $\mbox{ (5 marks)}$ | |
| 8. | Find the exact value of $\sin \frac{\pi}{12} \cos \frac{\pi}{6} \tan \frac{7 \pi}{12}$. $\mbox{ (5 marks)}$ | |
| 9. | Prove that $\operatorname{cosec} x-\frac{\sin x}{1+\cos x}=\cot x$. $\mbox{ (3 marks)}$ | |
| 10. | Prove that $1-\frac{\sin ^{2} x}{1+\cos x}=\cos x$. $\mbox{ (3 marks)}$ | |
| 11. | Prove the identity $\left(\frac{\sin 2 x}{\sin x}-\frac{\sin 2 x}{\cos x}\right)^{2}=4(1-\sin 2 x)$. $\mbox{ (3 marks)}$ | |
| 12. | Prove that $\frac{\cos x}{1+\sin x}=\sec x-\tan x$. $\mbox{ (5 marks)}$ | |
| 13. | Prove that $\frac{\tan \alpha+\tan \beta}{\tan \alpha-\tan \beta}=\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)} .$ By using this result calculate the value of the ratio $\sin 75^{\circ}: \sin 15^{\circ}$ in surd form. $\mbox{ (5 marks)}$ | |
| 14. | If $\alpha+\beta+\gamma=\pi$, show that $\sin \alpha+\sin \beta+\sin \gamma=4 \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2}$. $\mbox{ (5 marks)}$ | |
| 15. | Find the solutions, in the range $0 \leq x \leq 2 \pi$, of the equation $\cos x=-\frac{\sqrt{3}}{2}$. $\mbox{ (3 marks)}$ | |
| 16. | Solve the equation $2 \sin \theta \cos \theta-\sin \theta=0$ for $0^{\circ} \leq \theta \leq 180^{\circ}$. $\mbox{ (3 marks)}$ | |
| 17. | Solve the equation $2 \sin \theta \cos \theta-\cos \theta=0$ for $0^{\circ} \leq \theta \leq 180^{\circ}$. $\mbox{ (3 marks)}$ | |
| 18. | Solve the equation $\operatorname{cosec}^{2} x=2 \cot x$ for $0^{\circ} \leq x<360^{\circ}$. $\mbox{ (3 marks)}$ | |
| 19. | Solve the equation $\sec ^{2} x=2 \tan x$ for $0^{\circ} \leq x<360^{\circ}$. $\mbox{ (3 marks)}$ | |
| 20. | Solve the equation $\sin 2 x=\tan x$ for $0^{\circ} \leq x<360^{\circ}$. $\mbox{ (3 marks)}$ | |
| 21. | In $\triangle A B C a=9, b=11$ and $c=12$.Solve the triangle. $\mbox{ (5 marks)}$ | |
| 22. | Solve the triangle if $a=3, b=4, c=6$. $\mbox{ (5 marks)}$ | |
| 23. | In $\triangle A B C, a=5, b=8, c=7 .$ Solve the triangle. $\mbox{ (5 marks)}$ | |
| 24. | Solve $\triangle A B C$ with $a=9, b=11$ and $\gamma=60^{\circ}$. $\mbox{ (5 marks)}$ | |
| 25. | Solve the triangle $A B C$ with $a=9, b=10, \gamma=55^{\circ}$. $\mbox{ (5 marks)}$ | |
| 26. | In $\triangle A B C, b=5, c=11, \alpha=61^{\circ}$.Solve the triangle. $\mbox{ (5 marks)}$ | |
| 27. | Solve the triangle $A B C$ with $b=18.1, c=12.3$ and $\alpha=75^{\circ}$. $\mbox{ (5 marks)}$ | |
| 28. | Solve $\triangle A B C$ if $a=9, b=12$ and $\angle C=110^{\circ}$. $\mbox{ (5 marks)}$ | |
| 29. | In $\triangle A B C, a=5, \beta=75^{\circ}$ and $\gamma=41^{\circ}$.Find $\alpha, b$ and $c$. $\mbox{ (5 marks)}$ | |
| 30. | A town $\mathrm{P}$ is 25 miles away from the town $\mathrm{Q}$ in the direction $\mathrm{N} 35^{\circ} \mathrm{E}$ and a town $\mathrm{R}$ is 10 miles from $\mathrm{Q}$ in the direction $\mathrm{N} 42^{\circ} \mathrm{W}$. Calculate the distance and bearing of P from $R$. $\mbox{ (5 marks)}$ |
Answer (2011)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $\frac{31}{73}$ | |
| 2. | $\frac{5}{12}, \frac{5 \sqrt{26}}{26},-\frac{12}{13}$ | |
| 3. | $-\frac{2016}{4225}$ | |
| 4. | $\frac{77}{36},-\frac{13}{84}$ | |
| 5. | $\frac{240}{161}, \frac{76}{1443}$ | |
| 6. | $\frac{24}{25},-\frac{323}{325}$ | |
| 7. | $\frac{1}{2}$ | |
| 8. | $\frac{-3(\sqrt{2}+1)}{8}$ | |
| 9. | Prove | |
| 10. | Prove | |
| 11. | Prove | |
| 12. | Show | |
| 13. | $2+\sqrt{3}$ | |
| 14. | Show | |
| 15. | $\frac{5 \pi}{6}$ (or) $\frac{7 \pi}{6}$ | |
| 16. | $0^{\circ}$ (or) $60^{\circ}$ (or) $180^{\circ}$ | |
| 17. | $30^{\circ}$ (or) $90^{\circ}$ (or) $150^{\circ}$ | |
| 18. | $45^{\circ}$ (or) $225^{\circ}$ | |
| 19. | $45^{\circ}$ (or) $225^{\circ}$ | |
| 20. | $0^{\circ}$ (or) $45^{\circ}$ (or) $180^{\circ}$ (or) $315^{\circ}$ | |
| 21. | $45^{\circ} 49^{\prime}, 61^{\circ} 13^{\prime}, 72^{\circ} 58^{\prime}$ | |
| 22. | $26^{\circ} 23^{\prime}, 36^{\circ} 20^{\prime}, 117^{\circ} 17^{\prime} $ | |
| 23. | $38^{\circ} 12^{\prime}, 60^{\circ}, 81^{\circ} 48^{\prime}$ | |
| 24. | $10.15,50^{\circ} 10^{\prime}, 69^{\circ} 50^{\prime}$ | |
| 25. | $8.82,56^{\circ} 42^{\prime}, 68^{\circ} 18^{\prime}$ | |
| 26. | $9.627,27^{\circ} 1^{\prime}, 91^{\circ} 59^{\prime}$ | |
| 27. | $19.07,38^{\circ} 32^{\prime}, 66^{\circ} 28^{\prime}$ | |
| 28. | $17.29,40^{\circ} 43^{\prime}, 29^{\circ} 17^{\prime}$ | |
| 29. | $64^{\circ}$, 5.373, 3.65 | |
| 30. | 24.75 miles, N$58^{\circ} 15^{\prime}$E |
Group (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | If $0 < x < \frac{\pi}{2}$ and $\sin x=\frac{5}{13}$, find $\tan \left(\frac{\pi}{2}-x\right)$.$\text{ (3 marks)}$ | |
| 2. | Given that $\sin \beta=\frac{5}{13}, 90^{\circ} < \beta < 180^{\circ}$, find $\tan 2 \beta$.$\text{ (3 marks)}$ | |
| 3. | Given that $\cos 2 A=\frac{119}{169}$ and that $\angle A$ is acute, find without using tables, the value of $\cos A$ and $\sin A$.$\text{ (3 marks)}$ | |
| 4. | Solve the equation $\tan 2 x=\tan x$ for $0^{\circ} \leq x < 360^{\circ}$.$\text{ (3 marks)}$ | |
| 5. | Express $\cot 2 x$ in terms of $\cot x$.$\text{ (3 marks)}$ | |
| 6. | Find the values of $\theta, 0^{\circ} \leq \theta < 360^{\circ}$, which satisfy the equation : $2 \sin ^{2} \theta-\sin \theta=1$.$\text{ (5 marks)}$ | |
| 7. | Sove the equation $\cos 2 x=\sin x, 0 \leq x < 2 \pi$.$\text{ (5 marks)}$ | |
| 8. | Given that $\sin \alpha=\frac{3}{5}, \cos \beta=-\frac{12}{13}, \alpha$ and $\beta$ are in the same quadrant, find $\sin (\alpha+\beta), \cos (\alpha+\beta)$.$\text{ (5 marks)}$ | |
| 9. | If $\sin \theta=a$, where $\theta$ is an acute angle, express $\cot 2 \theta, \sec 2 \theta$ and $\operatorname{cosec} 2 \theta$ in terms of $a$.$\text{ (5 marks)}$ | |
| 10. | If $\sin \theta=a$, where $\theta$ is an acute angle, express $\sin 2 \theta, \tan 2 \theta$ and $\sin \frac{1}{2} \theta$ in terms of $a$.$\text{ (5 marks)}$ | |
| 11. | Without use of tables, evaluate $\tan (x+y+z)$ given that $\tan x=\frac{1}{3}, \tan y=\frac{2}{5}$ and $\tan z=\frac{3}{4}$.$\text{ (5 marks)}$ | |
| 12. | Solve $\triangle A B C$ with $a=9, b=10, c=15$.$\text{ (5 marks)}$ | |
| 13. | Solve the triangle $A B C$, if $a=20, b=7$ and $\gamma=40^{\circ}$.$\text{ (5 marks)}$ | |
| 14. | Solve $\triangle A B C$ with $a=6, b=10$ and $\gamma=110^{\circ}$.$\text{ (5 marks)}$ | |
| 15. | Solve $\triangle A B C$ with $\alpha=35^{\circ}, \beta=15^{\circ}, c=5$.$\text{ (5 marks)}$ | |
| 16. | In $\triangle A B C, \angle A=30^{\circ}, \angle B=85^{\circ}$, and $b=16$.Solve the triangle.$\text{ (5 marks)}$ | |
| 17. | Solve $\triangle A B C$ with $\beta=50^{\circ}, b=10, c=9$.$\text{ (5 marks)}$ | |
| 18. | To approximate the distance between two points $A$ and $B$ on opposite sides of a swamp, a surveyour selects a point $C$ and measures it to be 140 metres from $A$ and 260 metres from $B$.Then he measures the angle $A C B$, which turns out to be $49^{\circ}$.What is the distance from $A$ to $B$ ? $\text{ (5 marks)}$ | |
| 19. | A ship is $13 \mathrm{~km}$ from a boat in a direction $N 47^{\circ} E$ and a lighthouse is $15 \mathrm{~km}$ from that boat in a direction $S 25^{\circ} E$.Calculate the distance between the ship and the lighthouse.$\text{ (5 marks)}$ | |
| 20. | $A, B, C$ are three cities, $B$ is 20 miles from $A$ in a direction $N 47^{\circ} E .C$ is 27 miles away from $B$ in a direction $N 65^{\circ} W .$ Find the distance and direction of $A$ from $C$.$\text{ (5 marks)}$ | |
| 21. | A man travles $15 \mathrm{~km}$ in a direction $N 80^{\circ} E$ and then $5 \mathrm{~km}$ in a direction $N 40^{\circ} E$.What is his final distance and bearing from his starting point? $\text{ (5 marks)}$ | |
| 22. | A ship leaves harbour on a course $N 72^{\circ} E$, and after travelling for $50 \mathrm{~km}$, changes course to $108^{\circ}$.After a further $106 \mathrm{~km}$, find the distance of the ship from the harbour and its bearing from the harbour.$\text{ (5 marks)}$ | |
| 23. | Prove the identity: $(\operatorname{cosec} \theta-\cot \theta)^{2}\left(\frac{1+\cos \theta}{1-\cos \theta}\right)=1$.$\text{ (3 marks)}$ | |
| 24. | Prove the identity: $(\operatorname{cosec} \theta+\cot \theta)^{2}=\frac{1+\cos \theta}{1-\cos \theta}$.$\text{ (3 marks)}$ | |
| 25. | Prove that $\frac{\sin 3 x}{\sin x}+\frac{\cos 3 x}{\cos x}=4 \cos 2 x$.$\text{ (3 marks)}$ | |
| 26. | Prove that $\frac{\cos \theta+\sec \phi}{\cos \phi+\sec \theta}=\cos \theta \sec \phi$.$\text{ (3 marks)}$ | |
| 27. | Prove the identity $\tan \frac{\theta}{2}=\frac{1-\cos \theta}{\sin \theta}$.$\text{ (3 marks)}$ | |
| 28. | If $x+y+z=180^{\circ}$, show that $\sin (x+y)-\sin (y+z)=-2 \sin \frac{y}{2} \sin \frac{x-z}{2}$.$\text{ (3 marks)}$ | |
| 29. | Show that $\frac{\operatorname{cosec} \theta}{\sec 2 \theta}+\frac{\sec \theta}{\operatorname{cosec} 2 \theta}=\operatorname{cosec} \theta$.$\text{ (5 marks)}$ | |
| 30. | Show that $\cos 2 \theta \operatorname{cosec} \theta+\sin 2 \theta \sec \theta=\operatorname{cosec} \theta$.$\text{ (5 marks)}$ | |
| 31. | If $\delta+\lambda+\mu=360^{\circ}$, prove that $\tan \delta+\tan \lambda+\tan \mu=\tan \delta \tan \lambda \tan \mu$ Show also $$\cot \delta \cot \lambda+\cot \lambda \cot \mu+\cot \mu \cot \delta=1$$ $\text{ (5 marks)}$ | |
| 32. | If $\alpha+\beta+\gamma=\pi$, prove that $\sin (\alpha+\beta)+\sin (\beta+\gamma)=2 \cos \frac{\beta}{2} \cos \left(\frac{\alpha-\gamma}{2}\right)$.$\text{ (5 marks)}$ |
Answer (2010)
| $\quad\;\,$ | $\,$ | |
|---|---|---|
| 1. | $\frac{12}{5}$ | |
| 2. | $\frac{-120}{119}$ | |
| 3. | $\cos A=\frac{12}{13}, \sin A=\frac{5}{13}$ | |
| 4. | 0,180 | |
| 5. | $\cot 2 x=\frac{\cot ^{2} x-1}{2 \cot x}$ | |
| 6. | $90^{\circ}, 210^{\circ}, 330^{\circ}$ | |
| 7. | $\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$ | |
| 8. | $-\frac{56}{65} ; \frac{33}{65}$ | |
| 9. | $\frac{1-2 a^{2}}{2 a \sqrt{1-a^{2}}} ;\frac{1}{1-2 a^{2}} ; \frac{1}{2 a \sqrt{1-a^{2}}}$ | |
| 10. | $2 a \sqrt{1-a^{2}}, \frac{2 a \sqrt{1-a^{2}}}{1-2 a^{2}} ; \sqrt{\frac{1-\sqrt{1-a^{2}}}{2}}$ | |
| 11. | $\frac{83}{19}$ | |
| 12. | $\alpha=35^{\circ} 34^{\prime}, \beta=40^{\circ} 16^{\prime}, \gamma=104^{\circ} 10^{\prime}$ | |
| 13. | $c=15.32, \beta=17^{\circ} 5^{\prime}, \alpha=122^{\circ} 55^{\prime}$ | |
| 14. | $c=13.32, \alpha=25^{\circ} 8^{\prime}, \beta=44^{\circ} 52^{\prime}$ | |
| 15. | $\gamma=130^{\circ}, a=3.744, b=1.690 \quad$ | |
| 16. | $\gamma=65^{\circ}, a=8.032, c=14.55$ | |
| 17. | $\gamma=43^{\circ} 35^{\prime}, \alpha=86^{\circ} 25^{\prime}, a=13.02$ | |
| 18. | $198.6 \mathrm{~m}$ | |
| 19. | $22.68 \mathrm{~km}$ | |
| 20. | $26.92$ miles $; S 21^{\circ} 28^{\prime} E$ | |
| 21. | $19.10 \mathrm{~km} ; \mathrm{N} 70^{\circ} 19^{\prime} E$ | |
| 22. | $149.3 \mathrm{~km} ; S 83^{\circ} 20^{\prime} E$ | |
| 23. | Prove | |
| 24. | Prove | |
| 25. | Prove | |
| 26. | Prove | |
| 27. | Prove | |
| 28. | Prove | |
| 29. | Prove | |
| 30. | Prove | |
| 31. | Prove | |
| 32. | Prove |
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