Trigonometry (Part 2)

Trigonometry (Part 2)

6.9 Introduction

1. Find, giving your answers to $3$ decimal places

a. $\cot 304^{\circ}$

b. $\sec (-48^{\circ})$

c. cosec $62^{\circ}$

2. Using a calculator where necessary, find the values of the following, giving an non-exact answers correct to $3$ significant figures

a. $\sin \frac{2\pi}{5}$

b. $\sec \frac{\pi}{10}$

c. $\cot \frac{\pi}{12}$

d. cosec $\frac{17\pi}{6}$

e. $\cos \frac{7\pi}{8}$

f. $\tan \frac{5\pi}{12}$

g. $\sec (-\frac{11\pi}{12})$

h. $\cot (-\frac{\pi}{6})$

3. Find the exact values of

a. $\sec \frac{\pi}{4}$

b. cosec $\frac{\pi}{2}$

c. $\cot \frac{5\pi}{6}$

d. cosec $(-\frac{3\pi}{4})$

e. $\cot (-\frac{\pi}{3})$

f. $\sec \frac{13\pi}{6}$

g. $\cot (-\frac{5\pi}{2})$

h. $\sec \frac{7\pi}{6}$

4. Given that $\sin A=\frac{4}{5}$ and that $A$ is acute, without using a calculator, find the value of each of these

a. $\sec A$

b. cosec $A$

c. $\cot A$

5. Given that $\tan \theta=-\frac{1}{2}$ and $90^{\circ} < \theta < 360^{\circ}$, without using calculator, find the value of each of these

a. $\sec \theta$

b. cosec $\theta$

c. $\cot \theta$

6. Given that $\cos \theta=-\frac{1}{4}$ and $180^{\circ}<\theta <360^{\circ}$, without using a calculator, find the value of each of these

a. $\sec \theta$

b. cosec $\theta$

c. $\cot \theta$

7. Given that $\sec \theta=2$ and $0^{\circ}<\theta <180^{\circ}$, without using a calculator, find the value of each of these

a. $\sec \theta$

b. cosec $\theta$

c. $\cot \theta$

8. Given that cosec $B=3$ and that B is acute angle, without using a calculator, find the value of these

a. $\cos(90^{\circ}+B)$

b. $\tan(90^{\circ}-B)$

c. $\cot B$

9. Given that $\cot B=\frac{5}{2}$ and that $B$ is acute angle, without using a calculator, find the value of each these

a. $\sin B$

b. $\sin (90^{\circ}-B)$

c. $\sec (90^{\circ}-B)$

10. For $-360^{\circ} \le \theta \le 360^{\circ}$, sketch the graphs of the followings :

a. $y=\sin \theta$

b. $y=$ cosec $\theta$

c. $y=\cos \theta$

d. $y=\sec \theta$

e. $y=\tan \theta$

f. $y=\cot \theta$

6.10 Compound angles

Solve these following questions without using a calculator

1. Given that $A$ and $B$ are acute angles and that $\sin A=\frac{3}{5}$ and that $\cos B=\frac{5}{13}$, find the value of each of these

a. $\sin (A+B)$

b. $\cos (A-B)$

c. $\tan (A-B)

2. Given that $C$ and $D$ are acute angles and that $\cos C=\frac{12}{13}$ and that $\cos D=\frac{3}{5}$, find the value of each of these

a. $\cos (C+D)$

b. $\cos (C-D)$

c. $\cot (C+D)$

3. Given that $P$ and $Q$ are acute angles and that $\tan P=\frac{7}{24}$ and that $\tan Q=1$, find the value of each of these

a. $\sin (P+Q)$

b. $\cos (P+Q)$

c. $\tan (P-Q)$

4. Given that $\tan (A-B)=\frac{1}{2}$ and that $\tan A=3$, find the value of $\tan B$

5. Given that $\tan (P+Q)=5$ and that $\tan Q=2$, find the value of $\tan P$

6. Given that $\tan (\theta – 45^{\circ})=4$, find the value of $\tan \theta$

7. Given that $\tan (\theta +60^{\circ})=2$, find the value of $\cot \theta$

8. Given that $\cot (30^{\circ}-\theta)=3$, find the value of $\cot \theta$

9. Find the value of $\tan \theta$ for each of these:

a. $\sin (\theta – 30^{\circ})=\cos \theta$

b. $\sin (\theta + 45^{\circ})=\cos \theta$

c. $\cos (\theta + 60^{\circ})=\sin \theta$

d. $\sin (\theta + 60^{\circ})=\cos (\theta – 60^{\circ})$

e. $\cos (\theta + 60^{\circ})=2\cos (\theta + 30^{\circ})$

f. $\sin (\theta + 60^{\circ})=\cos (45^{\circ}-\theta)$

10. By writing $75$ as $30+45$, find the exact values of $\sin 75^{\circ}$ and $\tan 75^{\circ}$

11. Find the exact values of

a. $\cos 105^{\circ}$

b. $\sin 105^{\circ}$

c. $\tan 105^{\circ}$

12. Express $\cos (x+\frac{\pi}{3})$ in terms of $\cos x$ and $\sin x$

13. Use the expansions for $\sin (A+B)$ and $\cos (A+B)$ to simplify $\sin (\frac{3\pi}{2}+\theta)$ and $\cos (\frac{\pi}{2}+\theta)$

14. Express $\tan (\frac{\pi}{3}+x)$ and $\tan (\frac{5\pi}{6}-x)$ in terms of $\tan x$.

6.11 Double angles

1. Given that $\theta$ is an acute and that $\sin \theta = \frac{4}{5}$, find the value of each of these

a. $\sin 2\theta$

b. $\cos 2\theta$

c. $\tan 2\theta$

2. Given that $\theta$ is an acute and that $\cos \theta = \frac{5}{13}$, find the value of each of these

a. $\cos 2\theta$

b. cosec $2\theta$

c. $\cot 2\theta$

3. Given that $\theta$ is an acute and that $\tan \theta = 2$, find the value of each of these

a. $\tan 2\theta$

b. $\sin 2\theta$

c. $\sec 2\theta$

6.12 Half angle

1. Solve each of the following equations for $-360^{\circ} \le \theta \le 360^{\circ}$, giving your answers correct to one decimal place

a. $\sin \theta = \sin \frac{\theta}{2}$

b. $3\cos \frac{\theta}{2}=2\sin \theta$

c. $2\sin \theta = \tan \frac{\theta}{2}$

d. $2\cos \theta = 15\cos \frac{\theta}{2}+2$

2. By writing $\cos A$ in terms of $\frac{A}{2}$, find the alternative expression for $\frac{1-\cos A}{1+\cos A}$

3. If $\tan 2A=1$, find the possible values of $\tan A$. Hence state the exact value of $\tan 22\frac{1}{2}^{\circ}$

6.13 Mixed exercise

1. If $\frac{\sin (A-B)}{\sin (A+B)}=2$, find the value of $\tan A \cot B$

2. If $\tan 60^{\circ}=\sqrt{3}$, without using a calculator, show that $\tan 15^{\circ}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$

3. Prove that $2\sin (X+45^{\circ})\cos (X+45^{\circ})=\cos 2X$

4. Prove that $2\sin (Y+45^{\circ})\cos (Y-45^{\circ})=\cos 2Y$

5. If $\tan \theta = \frac{3}{4}$ and $180^{\circ}\le \theta \le 270^{circ}$, without using a calculator, find

a. $\tan 2\theta$

b. $\tan 3\theta$

c. $\tan \frac{1}{2}\theta$

6. If $P=Q+R$, prove that $\tan P-\tan Q-\tan R=\tan P\tan Q\tan R$

7. If $X+Y=45^{\circ}$, prove that $\tan X+\tan Y+\tan X\tan Y=1$

6.14 Trigonometric identities

1. Prove the following identities:

a. $\frac{1-\cos ^{2}A}{1-\sin ^{2}A}+\tan A\cot A=\sec ^{2}A$

b. $(\cot \theta + \tan \theta)\cos \theta =$ cosec$\theta$

c. $\cot \theta - \tan \theta = 2\cot 2\theta$

d. $cosec \theta - \sin \theta = \cos \theta \cot \theta$

2. Show that $\tan A =$ cosec $2A-\cot 2A$. Hence prove that $\tan 22.5^{\circ}=\sqrt{2}-1$

3. $(1-\sin x)\sec x = \frac{\cos x}{1+\sin x}$

4. $\cot x + \tan x=\frac{1}{\sin x \cos x}$

5. cosec $x- \sin x=\frac{\cos x}{\tan x}$

6. $(\sin \theta +\cos \theta)^{2}-1=2\sin \theta \cos \theta$

7. $(\sin \theta + \cos \theta)(1-\frac{1}{2}\sin 2\theta)=\sin ^{3}\theta + \cos ^{3}\theta$

8. $2\cos ^{3}\theta + \sin 2\theta \sin \theta = 2\cos \theta$

9. $\cos 4\theta – 4\cos 2\theta + 3 \equiv 8\sin ^{4}\theta$

6.15 Harmonic form [$a\sin \theta + b\cos \theta = R\sin (\theta + \alpha)$]

1. Find the following value of $R$ and $\tan \alpha$ in each of the following identities

a. $2\sin \theta + 4\cos \theta = R\sin (\theta + \alpha)$

b. $5\sin \theta - 12\cos \theta = R\sin (\theta - \alpha)$

c. $2\cos \theta + 5\sin \theta = R\cos (\theta - \alpha)$

d. $\cos \theta - \sin \theta = R\cos (\theta + \alpha)$

e. $5\sin 2\theta - 12\cos 2\theta = R\sin (2\theta - \alpha)$

f. $2\cos 2\theta + 5\sin 2\theta = R\cos (2\theta - \alpha)$

2. Find the greatest and least values of each of the following expressions, and state, correct to one decimal place, the smallest non-negative value of $\theta$ for which each occurs.

a. $12\sin \theta + 5\cos \theta$

b. $2\cos \theta + \sin \theta$

c. $7+3\sin \theta - 4\cos \theta$

d. $10-2\sin \theta + \cos \theta$

3. Solve each of the following equations for $0^{\circ}\le \theta \le 360^{\circ}$, giving your answers correct to one decimal place

a. $7\sin \theta – 4\cos \theta = 3$

b. $\cos \theta – 3\sin \theta = 2$

4. Express $3\cos x-4\sin x$ in the form $R\cos (x+\alpha)$ where $R>0$,  and $0^{\circ}<\alpha <90^{\circ}$. Hence

     a. Solve the equation $3\cos x-4\sin x = 2$, giving all the solution between 0 and 360.

     b. Find the greatest and least values, as $x$ varies, of the expression $\frac{1}{3\cos x-4\sin x+8}$

5. Given that $3\cos x-4\sin x=R\cos (x+\alpha)$, where $R>0$ and $0^{\circ}< \alpha < 90^{\circ}$, find the values of $R$ and $\alpha$. Hence solve the equation $3\cos 2\theta – 4\sin 2\theta =2$, for $0^{\circ}< \theta < 360^{\circ}$.

6. Express $5\sin \theta + 12\cos \theta$ in the form $r\sin (\theta + \alpha)$, where $r>0$ and $0^{\circ}< \alpha < 90^{\circ}$. Hence, find the maximum and minimum values of the expression $\frac{1}{5\sin \theta +12\cos \theta +15}$.