Trigonometry (Part 1)

Trigonometry (Part 1)

6.1 General angles

1. In each of the following cases find, correct to three decimal places, the value of (a)$\cos \theta$, (b) $\sin \theta$, (c) $\tan \theta$

i) 25

ii) 125

iii) 225

iv) 325

2. In each of the following cases find, correct to three decimal places, the value of

a. $\cos \frac{5}{7}\pi$

b. $\sin (-\frac{8}{5}\pi)$

c. $\tan \frac{9}{4}\pi$

d. $\tan (-\frac{5}{6}\pi)$

e. $\cos \frac{12}{7}\pi$

f. $\sin \frac{11}{5}\pi$

Express each of the following as trigonometric ratio of an acute angle, using the same trigonometric function as in the question

3. a. $\sin 200^{\circ}$

b. $\cos 240^{\circ}$

c. $\tan 160^{\circ}$

d. $\cos 310^{\circ}$

e. $\tan 220^{\circ}$

f. $\cos 490^{\circ}$

g. $\cos (-20^{\circ})$

h. $\cos (-280^{\circ})$

4. a. $\tan \frac{5}{8}\pi$

b. $\sin (-\frac{2}{5}\pi)$

c. $\cos \frac{7}{6}\pi$

d. $\sin \frac{17}{10}\pi$

e. $\cos (-\frac{8}{5}\pi)$

f. $\tan (-\frac{7}{4}\pi)$

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

a. $\sin \theta=0.3$

b. $\cos \theta=0.7$

c. $\tan \theta=2$

d. $\cos \theta=-0.5$

e. $\sin \theta=-0.35$

f. $\tan \theta=-7$

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

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

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

c. cosec $62^{\circ}$

7. Using a calculator where necessary, find the values of the following, giving any 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 $\frac{17\pi}{8}$

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

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

6.2 Sketch and use the graphs of trigonometry functions

1. i) Sketch and label, on the same diagram, the graphs of $y=2\sin x$ and $y=\cos 2x$, for interval $0\le x\le \pi$.

ii) Hence state the number of solutions of the equation $2\sin x=\cos 2x$ in the interval $0\le x\le \pi$.

2. The function $f$ is such that $f(x)=a-b\cos x$ for $0^{\circ} \le x \le 360^{\circ}$, where $a$ and $b$ are positive constants. The maximum value of $f(x)$ is $10$ and the minimum value is $-2$.

     i) Find the values of $a$ and $b$

     ii) Solve the equation $f(x)=0$

     iii) Sketch the graph of $y=f(x)$

3. The equation of a curve is $y=3 \cos 2x$. The equation of a line is $x+2y=\pi$. On the same diagram, sketch the curve and the line for $0\le x \le \pi$

4. i) Sketch the curve $y=2\sin x$ for $0\le x \le 2\pi$

ii) By adding a suitable straight line to your sketch, determine the number of real roots of the equation $2\pi \sin x=\pi -x$. State the equation of the straight line.

5. the function $f$ is defined by $f:x\mapsto 5-3\sin 2x$ for $0\le x \le \pi$

     i) Find the range of $f$

     ii) Sketch the graph of $y=f(x)$

     iii) State, with a reason, whether $f$ has an inverse

6.3 Special angles/Exact values

1. Without using a calculator, find the exact values of these trigonometric function

a. $\sin 135^{\circ}$

b. $\cos 120^{\circ}$

c. $\sin (-30^{\circ})$

d. $\tan 240^{\circ}$

e. $\cos 225^{\circ}$

f. $\tan (-330^{\circ})$

2. Without using calculator, find the exact values of:

a. $\tan \frac{5}{4}\pi$

b. $\sin \frac{2}{3}\pi$

c. $\cos \frac{7}{6}\pi$

d. $\sin \frac{4}{3}\pi$

e. $\cos \frac{7}{4}\pi$

f. $\tan \frac{5}{6}\pi$

3. Without using calculator, find the smallest positive angle which satisfies the following equations.

a. $\cos \theta ^{\circ}=\frac{1}{2}$

b. $\sin \beta ^{\circ}=-\frac{\sqrt{3}}{2}$

c. $\tan \theta ^{\circ}=-\sqrt{3}$

d. $\cos \beta ^{\circ}=\frac{\sqrt{3}}{2}$

e. $\tan \theta ^{\circ}=-\frac{\sqrt{3}}{3}$

f. $\tan \beta ^{\circ}=-1$

4. Find the exact value of $\cos 45^{\circ}+\cos 135^{\circ}+\cos 225^{\circ}+\cos 315^{\circ}$

5. Find the exact value of $\tan 30^{\circ}+\tan 120^{\circ}$

6.4 Area of triangle [$\Delta = \frac{1}{2}ab \sin C$]

Find the area of the $\Delta ABC$, if it is given that

1. $A=48^{\circ},b=6cm$ and $c=5cm$

2. $B=70^{\circ},a=15cm$ and $c=5cm$

3. $C=43^{\circ},a=11cm$ and $c=9cm$

4. Calculate the largest angle and the shortest height of a triangle which has sides of lengths $6cm$, $7cm$ and $10cm$

5. In a triangle $ABC$, $BC=20cm$, $B=30^{\circ}$ and $C=45^{\circ}$. Find the area of this triangle.

6.5 Use the notations $\sin ^{-1}(x)$, $\cos ^{-1}(x)$ and $\tan ^{-1}(x)$

Without using a calculator, find the smallest positive value of the followings in terms of degrees:

1. a. $\cos ^{-1}\frac{1}{2}$

b. $\sin ^{-1}\frac{\sqrt{3}}{2}$

c. $\tan ^{-1}\sqrt{3}$

d. $\cos ^{-1}(-\frac{\sqrt{3}}{2})$

e. $\tan ^{-1}(-\frac{\sqrt{3}}{3})$

f. $\tan ^{-1}(-1)$

Without using a calculator, find the smallest positive value of the followings in terms of radians:

2. a. $\cos ^{-1}\frac{\sqrt{3}}{2}$

b. $\sin ^{-1}\frac{1}{2}$

c. $\tan ^{-1}\frac{\sqrt{3}}{2}$

d. $\cos ^{-1}(-\frac{\sqrt{3}}{2})$

e. $\tan ^{-1}(-\sqrt{3})$

f. $\sin ^{-1}(-1)$

3. Find

a. $\cos(\cos ^{-1}0.5)$

b. $\sin(\sin ^{-1}\frac{\sqrt{3}}{2})$

c. $\tan(\tan ^{-1}\sqrt{3})$

4. Given that $x=\sin ^{-1}(\frac{2}{5})$, find the exact value of

i) $\cos ^{2}x$

ii) $\tan ^{2}x$

6.6 Solutions of trigonometric equations

1. Find all the angles between $0^{\circ}$ and $360^{\circ}$ which satisfy the equation, giving your answers correct to one decimal place

a. $10\sin x-7=0$

b. $5\cos x+2=0$

c. $1+2\cos x=0$

d. $7(2\tan x -3)=4(\tan x+2)$

e. $\sin (x+20^{\circ})=0.4$

f. $\cos (x-50^{\circ})=-0.3$

2. Find all the angles $x$, where $-180^{\circ}< x<180^{\circ}$, such that

a. $\cos x=0.75$

b. $\sin x=-\frac{1}{2}$

c. $\tan x=2$

d. $10\cos x + 4=0$

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

3. $\sin x \cos x=2\cos x$

4. $3\sin x \cos x=\sin x$

5. $\cos x-3\sin x=0$

6. $\cos ^{2}x=\frac{3}{4}$

7. $\sin ^{2}x=\frac{1}{9}$

8. $\tan ^{2}x=\frac{9}{4}$

9. $2\tan ^{2}x + 1=3\tan x$

10. $4\cos ^{2}x-5\cos x+1=0$

11. $3\sin ^{2}x-5\sin x+2=0$

12. Find all values of $x$, where $0^{\circ} < x < 360^{\circ}$, which satisfy the equation $\tan x +\frac{4}{\tan x}=\frac{4}{\cos x}$

6.7 Trigonometric Identities Proving

1. Prove that $\frac{\cos ^{2}x}{1-\sin x}=1+\sin x$

2. $\sin x = \cos x \tan x$

3. $\cos ^{4}\theta - \sin ^{4}\theta = \cos ^{2}\theta - \sin ^{2}\theta$

4. $1-\frac{\sin ^{2}x}{1+\cos x}=\cos x$

5. $1+\frac{\cos ^{2}x}{\sin x-1}=-\sin x$

6. $\sin x +\cos x=\frac{1-2\cos ^{2}x}{\sin x - \cos x}$

7. $\tan ^{2}x-\sin ^{2}x=\tan ^{2}x \sin ^{2}x$

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

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

10. $\frac{1+\sin x+\cos x}{\cos x}=\frac{1-\sin x +\cos x}{1-\sin x}$

11. $\frac{1}{\sin x}-\sin x=\frac{\cos x}{\tan x}$

12.  $\frac{1-\sin x}{\cos x}=\frac{\cos x}{1+\sin x}$

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

14. $2\cos ^{3}\theta +2\sin ^{2}\theta \cos \theta = 2\cos \theta$

15. Prove the identity $\tan x+\frac{\cos x}{\sin x}\equiv \frac{1}{\sin x}\cos x$

16. Express the equation $3(2\sin x-\cos x)=2(\sin x-3\cos x)$ in terms of $\tan x$

17. Show that $(\sin \theta + \cos \theta)(1-\sin \theta \cos \theta)=\sin ^{3}\theta +\cos ^{3}\theta$. 

Hence, for $0^{\circ} \le x \le 360^{\circ}$, solve the equation $(\sin x + \cos x)(1-\sin x \cos x)=9\sin ^{3}x$

18. Prove the identity $\frac{\cos ^{2}\theta}{\sin \theta (1-\sin \theta)} \equiv 1+\frac{1}{\sin \theta}$. Hence solve the equation $\frac{\cos ^{2}\theta}{\sin \theta (1-\sin \theta)}=4$, for $0^{\circ} \le x \le 360^{\circ}$.

19. Express the equation $2\tan ^{2}\theta \sin ^{2}\theta = 1$ in terms of $\sin x$. Hence, solve the equation  $2\tan ^{2}\theta \sin ^{2}\theta = 1$ for [0^{\circ},360^{\circ}]

6.8 Mixed Exercise

1. Find all the angles between $0^{\circ}$ and $360^{\circ}$ which satisfy the equation $3\cos x=8\tan x$

2. If $4 \le \theta \le 6$, find the value of $\theta$ for which $2\cos (\frac{2\theta}{3})+\sqrt{3}=0$.

3. Solve, for $0^{\circ} \le x \le 360^{\circ}$, the equation $2\frac{\cos x}{\sin x}=1+\tan x$

4. Given that $x$ is measured in radians, find the two smallest positive values of $x$ such that $6\sin (2x+1)+5=0$

5. Solve the equation $3\sin (\frac{x}{2}-1)=1$ for $0<x<6\pi$ radians.