# Trigonometry (Myanmar Exam Board)

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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)

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) $\cos(A+B)=\frac{\sqrt{10}-2\sqrt{35}}{15}$ $\cos(2A-B)=\frac{3\sqrt 2-4\sqrt 7}{15}$

23.Prove
24.Prove
25.Prove
26.Prove
27.Prove
28.Prove
29.Prove
30.Prove
31.Prove
32.Prove