• Home
  • Textbooks
  • Higher Engineering Mathematics
  • Introduction to trigonometry

Higher Engineering Mathematics

John Bird

Chapter 12

Introduction to trigonometry - all with Video Answers

Educators

Chapter Questions

02:14

Problem 1

In Fig. 12.2, find the length of $E F$.
Figure $12.2$
By Pythagoras' theorem:
Hence
$$
\begin{aligned}
e^{2} &=d^{2}+f^{2} \\
13^{2} &=d^{2}+5^{2} \\
169 &=d^{2}+25 \\
d^{2} &=169-25=144 \\
d &=\sqrt{144}=12 \mathrm{~cm} \\
E F &=12 \mathrm{~cm}
\end{aligned}
$$
Thus.
i.e.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:04

Problem 2

Two aircraft leave an airfield at the same time. One travels due north at an average speed of $300 \mathrm{~km} / \mathrm{h}$ and the other due west at an average speed of $220 \mathrm{~km} / \mathrm{h}$. Calculate their distance apart after four hours.

After four hours, the first aircraft has travelled $4 \times 300=$ $1200 \mathrm{~km}$, due north, and the second aireraft has travelled $4 \times 220=880 \mathrm{~km}$ due west, as shown in Fig. 12.3. Distance apart after four hours $=B C$.
Figure $12.3$
From Pythagoras' theorem:
$$
B C^{2}=1200^{2}+880^{2}=1440000+774400
$$
and $B C=\sqrt{(2214400)}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:47

Problem 3

If $\cos X=\frac{9}{41}$ determine the value of the other five trigonometry ratios.
Fig. $12.7$ shows a right-angled triangle $X Y Z$. Since $\cos X=\frac{9}{41}$, then $X Y=9$ units and $X Z=41$ units.
Using Pythagoras' theorem: $41^{2}=9^{2}+Y Z^{2}$ from which $Y Z=\sqrt{\left(41^{2}-9^{2}\right)}=40$ units.
Thus
$$
\begin{aligned}
\sin X &=\frac{40}{41}, \tan X=\frac{40}{9}=4 \frac{4}{9}, \\
\operatorname{cosec} X &=\frac{41}{40}=1 \frac{1}{40} \\
\sec X &=\frac{41}{9}=4 \frac{5}{9} \text { and } \cot X=\frac{9}{40}
\end{aligned}
$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:37

Problem 4

If $\sin \theta=0.625$ and $\cos \theta=0.500$ determine, without using trigonometric tables or calculators, the values of $\operatorname{cosec} \theta, \sec \theta, \tan \theta$ and $\cot \theta$.
$$
\begin{aligned}
\operatorname{cosec} \theta &=\frac{1}{\sin \theta}=\frac{1}{0.625}=1.60 \\
\sec \theta &=\frac{1}{\cos \theta}=\frac{1}{0.500}=\mathbf{2 . 0 0} \\
\tan \theta &=\frac{\sin \theta}{\cos \theta}=\frac{0.625}{0.500}=\mathbf{1 . 2 5} \\
\cot \theta &=\frac{\cos \theta}{\sin \theta}=\frac{0.500}{0.625}=\mathbf{0 . 8 0}
\end{aligned}
$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:17

Problem 5

Point $A$ lies at co-ordinate $(2,3)$ and point $B$ at $(8,7)$. Determine (a) the distance $A B$, (b) the gradient of the straight line $A B$, and (c) the angle $A B$ makes with the horizontal.
(a) Points $A$ and $B$ are shown in Fig. $12.8$ (a).
In Fig. $12.8(\mathrm{~b})$, the horizontal and vertical lines $A C$ and $B C$ are constructed.

Since $A B C$ is a right-angled triangle, and $A C=(8-2)=6$ and $B C=(7-3)=4$, then by Pythagoras' theorem
$$
A B^{2}=A C^{2}+B C^{2}=6^{2}+4^{2}
$$
and $A B=\sqrt{\left(6^{2}+4^{2}\right)}=\sqrt{52}=7.211$, correct to 3 decimal places.
(b) The gradient of $A B$ is given by $\tan A$, i.e. gradient $=\tan A=\frac{B C}{A C}=\frac{4}{6}=\frac{2}{3}$
(c) The angle $A B$ makes with the horizontal is given by $\tan ^{-1} \frac{2}{3}=33.69^{\circ}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
00:48

Problem 6

Determine, correct to 4 decimal places, $\sin 43^{\circ} 39^{\prime}$
$$
\begin{aligned}
\sin 43^{\circ} 39^{\prime}=\sin 43 \frac{39}{60} &=\sin 43.65^{\circ} \\
&=0.6903
\end{aligned}
$$
This answer can be obtained using the calculator as follows:
1. Press $\sin$.
2. Enter 43
3. Press $^{\circ}$ ?
4. Enter 39
5. Press
6. Press )
7. Press $=$ Answer $=\mathbf{0 . 6 9 0 2 5 1 2} \ldots$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
00:47

Problem 7

Determine, correct to 3 decimal places, $6 \cos 62^{\circ} 12^{\prime}$
$$
\begin{aligned}
6 \cos 62^{\circ} 12^{\prime}=6 \cos 62 \frac{12^{\circ}}{60} &=6 \cos 62.20^{\circ} \\
&=2.798
\end{aligned}
$$
This answer can be obtained using the calculator as follows:
1. Enter 6
2. Press $\cos$
3. Enter 62
4. Press $^{\circ} \%$
5. Enter 12 .
6. Press $^{\circ}$ "
7. Press)
8. Press $=$ Answer $=2.798319 \ldots$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:00

Problem 8

Evaluate, correct to 4 decimal places:
(a) sine $168^{\circ} 14^{\prime}$ (b) cosine $271.41^{\circ}$
(c) tangent $98^{\circ} 4^{\prime}$
(a) sine $168^{\circ} 14^{\prime}=$ sine $168 \frac{14^{\circ}}{60}=0.2039$
(b) $\quad \operatorname{cosine} 271.41^{\circ}=\mathbf{0 . 0 2 4 6}$
(c) tangent $98^{\circ} 4^{\prime}=\tan 98 \frac{4^{\circ}}{60}=-7.0558$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:38

Problem 9

Evaluate, correct to 4 decimal places:
(a) secant $\mid 61$
(b) secant $302^{\circ} 29^{\prime}$
(a) $\sec 161^{\circ}=\frac{1}{\cos 161^{\circ}}=-\mathbf{1 . 0 5 7 6}$
(b) $\sec 302^{\circ} 29^{\prime}=\frac{1}{\cos 302^{\circ} 29^{\prime}}=\frac{1}{\cos 302 \frac{29^{\circ}}{60}}$ $=\mathbf{1 . 8 6 2 0}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:29

Problem 10

Evaluate, correct to 4 significant figures:
(a) cosecant $279.16$
(b) cosecant $49^{\circ} 7^{\prime}$
(a) $\operatorname{cosec} 279.16^{\circ}=\frac{1}{\sin 279.16^{\circ}}=-\mathbf{1 . 0 1 3}$
(b) $\quad \operatorname{cosec} 49^{\circ} 7^{\prime}=\frac{1}{\sin 49^{\circ} 7^{\prime}}=\frac{1}{\sin 49 \frac{7^{\circ}}{60}}$ $=\mathbf{1 . 3 2 3}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:16

Problem 11

. Evaluate, correct to 4 decimal places:
(a) cotangent $17.49^{\circ}$
(b) cotangent $163^{\circ} 52^{\prime}$
(a) $\cot 17.49^{\circ}=\frac{1}{\tan 17.49^{\circ}}=3.1735$
(b) $\cot 163^{\circ} 52^{\prime}=\frac{1}{\tan 163^{\circ} 52^{\prime}}=\frac{1}{\tan 163 \frac{52^{\circ}}{60}}$ $=-3.4570$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:01

Problem 12

Evaluate, correct to 4 significant figures:
(a) $\sin 1.481$
(b) $\cos (3 \pi / 5)$
(c) $\tan 2.93$ (a) $\sin 1.481$ means the sine of $1.481$ radians. Hence a calculator needs to be on the radian function. Hence $\sin 1.481=\mathbf{0 . 9 9 6 0}$
(b) $\cos (3 \pi / 5)=\cos 1.884955 \ldots=-0.3090$
(c) $\tan 2.93=-0.2148$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:02

Problem 13

Evaluate, correct to 4 decimal places:
(a) secant $5.37$
(b) cosecant $\pi / 4$
(c) cotangent $\pi / 24$
(a) Again, with no degrees sign, it is assumed that $5.37$ means $5.37$ radians.
Hence $\sec 5.37=\frac{1}{\cos 5.37}=1.6361$
(b) $\operatorname{cosec}(\pi / 4)=\frac{1}{\sin (\pi / 4)}=\frac{1}{\sin 0.785398 \ldots}$ $=1.4142$
(c) $\cot (5 \pi / 24)=\frac{1}{\tan (5 \pi / 24)}=\frac{1}{\tan 0.654498 \ldots}$ $=1.3032$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:04

Problem 14

Evaluate, correct to 4 decimal places:
(a) secant $5.37$
(b) cosecant $\pi / 4$
(c) cotangent $\pi / 24$
(a) Again, with no degrees sign, it is assumed that $5.37$ means $5.37$ radians.
Hence $\sec 5.37=\frac{1}{\cos 5.37}=1.6361$
(b) $\operatorname{cosec}(\pi / 4)=\frac{1}{\sin (\pi / 4)}=\frac{1}{\sin 0.785398 \ldots}$ $=1.4142$
(c) $\cot (5 \pi / 24)=\frac{1}{\tan (5 \pi / 24)}=\frac{1}{\tan 0.654498 \ldots}$ $=1.3032$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:46

Problem 15

Find, in degrees, the acute angle $\sin ^{-1} 0.4128$ correct to 2 decimal places.
$\sin ^{-1} 0.4128$ means 'the angle whose
sine is $0.4128$
Using a calculator:
1. Press shift 2. Press $\sin$ 3. Fnter $0.4128$
4. Press ) 5. Press $=$ The answer $24.380848 \ldots$
is displayed
$\sin ^{-1} 0.4128=24.38^{\circ}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:34

Problem 16

Find the acute angle $\tan ^{-1} 7.4523$ in degrees and minutes.
$\tan ^{-1} 7.4523$ means 'the angle whose
tangent is $7.4523^{\prime}$
Jsing a calculator:
Press shift 2 . Press tan 3. Enter 7.4523
Press ) $5 .$ Press $=$ The answer $82.357318 \ldots$ is displayed
Hen tar $-19.17$
correct to the nearest minute

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:22

Problem 17

Determine the acute angles:
(a) $\sec ^{-1} 2.3164$
(b) $\operatorname{cosec}^{-1} 1.1784$
(c) $\cot ^{-1} 2.1273$
(a)
$$
\begin{aligned}
\sec ^{-1} 2.3164 &=\cos ^{-1}\left(\frac{1}{2.3164}\right) \\
&=\cos ^{-1} 0.4317 \ldots \\
&=\mathbf{6 4 . 4 2 ^ { \circ }} \text { or } 64^{\circ} \mathbf{2 5}^{\prime}
\end{aligned}
$$
or $1.124$ radians
$$
\text { (b) } \begin{aligned}
\operatorname{cosec}^{-1} 1.1784 &=\sin ^{-1}\left(\frac{1}{1.1784}\right) \\
&=\sin ^{-1} 0.8486 \ldots \\
&=58.06^{\circ} \text { or } 58^{\circ} 4^{\prime}
\end{aligned}
$$
or $1.013$ radians
$$
\text { (c) } \begin{aligned}
\cot ^{-1} 2.1273 &=\tan ^{-1}\left(\frac{1}{2.1273}\right) \\
&=\tan ^{-1} 0.4700 \ldots \\
&=\mathbf{2 5 . 1 8}^{\circ} \text { or } \mathbf{2 5}^{\circ} \mathbf{1 1}^{\prime}
\end{aligned}
$$
or $\mathbf{0 . 4 3 9}$ radians

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:10

Problem 18

Evaluate the following expression, correct to 4 significant figures:
$$
\frac{4 \sec 32^{\circ} 10^{\prime}-2 \cot 15^{\circ} 19^{\prime}}{3 \operatorname{cosec} 63^{\circ} 8^{\prime} \tan 14^{\circ} 57^{\prime}}
$$ By calculator:
$$
\begin{aligned}
\sec 32^{\circ} 10^{\prime} &=1.1813, \cot 15^{\circ} 19^{\prime}=3.6512 \\
\operatorname{cosec} 63^{\circ} 8^{\prime} &=1.1210, \tan 14^{\circ} 57^{\prime}=0.2670
\end{aligned}
$$
$$
\text { Hence } \begin{aligned}
& \frac{4 \sec 32^{\circ} 10^{\prime}-2 \cot 15^{\circ} 19^{\prime}}{3 \operatorname{cosec} 63^{\circ} 8^{\prime} \tan 14^{\circ} 57^{\prime}} \\
&=\frac{4(1.1813)-2(3.6512)}{3(1.1210)(0.2670)} \\
&=\frac{4.7252-7.3024}{0.8979} \\
&=\frac{-2.5772}{0.8979}=-2.870
\end{aligned}
$$
correct to 4 significant figures.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:15

Problem 19

Fvaluate correct to 4 decimal places:
(a) $\sec \left(-115^{\circ}\right)$
(b) $\operatorname{cosec}\left(-95^{\circ} 47^{\prime}\right)$
(a) Positive angles are considered by convention to be anticlockwise and negative angles as clockwise. Hence $-115^{\circ}$ is actually the same as $245^{\circ}$ (i.e. $\left.360^{\circ}-115^{\circ}\right)$
Hence
$$
\begin{aligned}
\sec \left(-115^{\circ}\right) &=\sec 245^{\circ}=\frac{1}{\cos 245^{\circ}} \\
&=-\mathbf{2 . 3 6 6 2}
\end{aligned}
$$
(b) $\operatorname{cosec}\left(-95^{\circ} 47^{\prime}\right)=\frac{1}{\sin \left(-95 \frac{47^{\circ}}{60}\right)}=-\mathbf{1 . 0 0 5 1}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:07

Problem 20

In triangle $E F G$ in Fig. $12.11$, calculate angle $G$. With reference to $\angle G$, the two sides of the triangle given are the opposite side $E F$ and the hypotenuse $E G ;$ hence, sine is used,
i.e. $\quad \sin G=\frac{2.30}{8.71}=0.26406429 \ldots$
from which, $\quad G=\sin ^{-1} 0.26406429 \ldots$
i.e. $G=15.311360 \ldots$
Hence, $\quad \angle \mathbf{G}=\mathbf{1 5 . 3 1}^{\circ}$ or $\mathbf{1 5}^{\circ} \mathbf{1 9}^{\prime}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:48

Problem 21

In triangle $P Q R$ shown in Fig. 12.14, find the lengths of $P Q$ and $P R$. $\tan 38^{\circ}=\frac{P Q}{Q R}=\frac{P Q}{7.5}$
hence $\quad P Q=7.5 \tan 38^{\circ}=7.5(0.7813)$
$=5.860 \mathrm{~cm}$
$\cos 38^{\circ}=\frac{Q R}{P R}=\frac{7.5}{P R}$
hence $\quad P R=\frac{7.5}{\cos 38^{\circ}}=\frac{7.5}{0.7880}=9.518 \mathrm{~cm}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:58

Problem 22

Solve the triangle $A B C$ shown in Fig. $12.15$. To 'solve triangle $A B C$ ' means 'to find the length $A C$ and angles $B$ and $C$ '
$$
\sin C=\frac{35}{37}=0.94595
$$
hence $\angle C=\sin ^{-1} 0.94595=71.08^{\circ}=71^{\circ} 5^{\prime}$ $\angle B=180^{\circ}-90^{\circ}-71^{\circ} 5^{\prime}=18^{\circ} 55^{\prime}$ (since angles in a triangle add up to $180^{\circ}$ )
$$
\sin B=\frac{A C}{37}
$$
hence
$$
\begin{aligned}
A C &=37 \sin 18^{\circ} 55^{\prime}=37(0.3242) \\
&=12.0 \mathrm{~mm}
\end{aligned}
$$
or, using Pythagoras' theorem, $37^{2}=35^{2}+A C^{2}$, from which, $A C=\sqrt{\left(37^{2}-35^{2}\right)}=12.0 \mathrm{~mm}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:23

Problem 23

Solve triangle $X Y Z$ given $\angle X=90^{\circ}, \angle Y=23^{\circ} 17^{\prime}$ and $Y Z=20.0 \mathrm{~mm}$. Determine also its area.It is always advisable to make a reasonably accurate sketch so as to visualise the expected magnitudes of unknown sides and angles. Such a sketch is shown in Fig. 12,16 .
$$
\begin{aligned}
\angle Z &=180^{\circ}-90^{\circ}-23^{\circ} 17^{\prime}=66^{\circ} 43^{\prime} \\
\sin 23^{\circ} 17^{\prime} &=\frac{X Z}{20.0}
\end{aligned}
$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:09

Problem 24

An electricity pylon stands on horizontal ground. At a point $80 \mathrm{~m}$ from the base of the pylon, the angle of elevation of the top of the pylon is $23^{\circ}$. Calculate the height of the pylon to the nearest metre.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
05:34

Problem 25

A surveyor measures the angle of elevation of the top of a perpendicular building as $19^{\circ}$. He moves $120 \mathrm{~m}$ nearer the building and finds the angle of elevation is now $47^{\circ}$. Determine the height of the building.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
05:10

Problem 26

The angle of depression of a ship viewed at a particular instant from the top of a $75 \mathrm{~m}$ vertical cliff is $30^{\circ}$. Find the distance of the ship from the base of the cliff at this instant. The ship is sailing away from the cliff at constant speed and one minute later its angle of depression from the top of the cliff is $20^{\circ}$. Determine the speed of the ship in $\mathrm{km} / \mathrm{h}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:13

Problem 27

In a triangle $X Y Z, \angle X=51^{\circ}$ $\angle Y=67^{\circ}$ and $Y Z=15.2 \mathrm{~cm}$. Solve the triangle and find its area.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
06:25

Problem 28

Solve the triangle $P Q R$ and find its area given that $Q R=36.5 \mathrm{~mm}, P R=29.6 \mathrm{~mm}$ and $\angle Q=36^{\circ}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:59

Problem 29

Solve triangle $D E F$ and find its area given that $E F=35.0 \mathrm{~mm}, D E=25.0 \mathrm{~mm}$ and $\angle E=64^{\circ}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
06:54

Problem 30

A triangle $A B C$ has sides $a=9.0 \mathrm{~cm}, b=7.5 \mathrm{~cm}$ and $c=6.5 \mathrm{~cm}$. Determine its three angles and its area.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:25

Problem 31

. A room $8.0 \mathrm{~m}$ wide has a span roof which slopes at $33^{\circ}$ on one side and $40^{\circ}$ on the other. Find the length of the roof slopes, correct to the nearest centimetre.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:59

Problem 32

Two voltage phasors are shown in Fig. 12.31. If $V_{1}=40 \mathrm{~V}$ and $V_{2}=100 \mathrm{~V}$ determinethe value of their resultant (i.e. length $O A$ ) and the angle the resultant makes with $V_{1}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:15

Problem 33

In Fig. $12.32, P R$ represents the inclined jib of a crane and is $10.0$ long. $P Q$ is $4.0 \mathrm{~m}$ long. Determine the inclination of the jib to the vertical and the length of tie $Q R$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:38

Problem 34

A vertical aerial stands on horizontal ground. A surveyor positioned due east of the aerial measures the elevation of the top as $48^{\circ}$. He moves due south $30.0 \mathrm{~m}$ and measures the elevation as $44^{\circ}$. Determine the height of the aerial.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:28

Problem 35

A crank mechanism of a petrol engine is shown in Fig. 12.37. Arm $O A$ is $10.0 \mathrm{~cm}$ long and rotates clockwise about $\mathrm{O}$. The connecting $\operatorname{rod} A B$ is $30.0 \mathrm{~cm}$ long and end $B$ is constrained to move horizontally.
Figure 12.37]
(a) For the position shown in Fig. $12.37$ determine the angle between the connecting $\operatorname{rod} A B$ and the horizontal and the length of $O B$.
(b) How far does $B$ move when angle $A O B$ changes from $50^{\circ}$ to $120^{\circ}$ ?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:58

Problem 36

The area of a field is in the form of a quadrilateral $A B C D$ as shown in Fig. $12.39$.
Determine its area.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator