“JUST THE MATHS”
UNIT NUMBER
3.1
TRIGONOMETRY 1
(Angles & trigonometric functions)
by
A.J.Hobson
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5

Introduction
Angular measure
Trigonometric functions
Exercises
Answers to exercises


UNIT 3.1 - TRIGONOMETRY 1
ANGLES AND TRIGONOMETRIC FUNCTIONS
3.1.1 INTRODUCTION
The following results will be assumed without proof:
(i) The Circumference, C, and Diameter, D, of a circle are directly proportional to each
other through the formula
C = πD
or, if the radius is r,
C = 2πr.
(ii) The area, A, of a circle is related to the radius, r, by means of the formula
A = πr2 .

is, 2π radians is equivalent to 360◦ or, in other words π radians is equivalent to 180◦ .
(ii) In the diagram overleaf, the arclength from A to B will be given by
θ
× 2πr = rθ,
2π
assuming that θ is measured in radians.
1


(iii) In the diagram below, the area of the sector ABC is given by
θ
1
× πr2 = r2 θ.
2π
2

A

C

✡❅
✡❅
❅
❅
❅
✡❅
❅❅❅
❅
❅
✡❅

× 30 = π6 .

3. 45◦ is equivalent to

π
180

× 45 = π4 .

4. 60◦ is equivalent to

π
180

× 60 = π3 .

5. 75◦ is equivalent to

π
180

× 75 =

6. 90◦ is equivalent to

π
180

× 90 = π2 .


“negative”.
3.1.3 TRIGONOMETRIC FUNCTIONS
We first consider a right-angled triangle in one corner of which is an angle θ other than
the right-angle itself. The sides of the triangle are labelled in relation to this angle , θ, as
“opposite”, “adjacent” and “hypotenuse” (see diagram below).

✟
✟

hypotenuse✟✟

opposite

✟
✟✟
✟
✟✟ adjacent

✟✟θ

For future reference, we shall assume, without proof, the result known as “Pythagoras’
Theorem”. This states that the square of the length of the hypotenuse is equal to the sum
of the squares of the lengths of the other two sides.
DEFINITIONS
(a) The “sine” of the angle θ, denoted by sin θ, is defined by
sin θ ≡

opposite
;
hypotenuse

✟ θ

✟(x, y)
✟✟
h ✟✟
✟
✟✟
✲x

O

For any values of x and y, positive, negative or zero, the three basic trigonometric functions
are defined in general by the formulae
y
sin θ ≡ ;
h
x
cos θ ≡ ;
h
y
sin θ
tan θ ≡ ≡
.
x
cos θ
Clearly these reduce to the original definitions in the case when θ is a positive acute angle.
Trigonometric functions can also be called “trigonometric ratios”.
(iii) It is useful to indicate diagramatically which of the three basic trigonometric functions
have positive values in the various quadrants.


cot θ ≡

1
.
tan θ

(v) The values of the functions sin θ, cos θ and tan θ for the particluar angles 30◦ , 45◦ and
60◦ are easily obtained without calculator from the following diagrams:

√

✁❆
✁ ❆
✁
❆
✁30◦ 30◦❆
✁
❆
✁
❆
❆ 2
2✁
✁
❆
✁
❆
✁
❆
√
✁

1

1

1

The diagrams show that
(a) sin 45◦ =

√1 ;
2

(d) sin 30◦ = 12 ;
(g) sin 60◦ =

√

3
;
2

(b) cos 45◦ =

√1 ;
2
√

(c) tan 45◦ = 1;

3


3. A wheel is turning at the rate of 48 revolutions per minute. Express this angular speed
in
(a) revolutions per second; (b) radians per minute; (c) radians per second.

5


4. A wheel, 4 metres in diameter, is rotating at 80 revolutions per minute. Determine the
distance, in metres, travelled in one second by a point on the rim.
5. A chord AB of a circle, radius 5cms., subtends a right-angle at the centre of the circle.
Calculate, correct to two places of decimals, the areas of the two segments into which
AB divides the circle.
6. If tan θ is positive and cos θ = − 45 , what is the value of sin θ ?
7. Determine the length of the chord of a circle, radius 20cms., subtending an angle of
150◦ at the centre.
8. A ladder leans against the side of a vertical building with its foot 4 metres from the
building. If the ladder is inclined at 70◦ to the ground, how far from the ground is the
top of the ladder and how long is the ladder ?
3.1.5 ANSWERS TO EXERCISES
1. (a)

13π
;
36

(b)

7π
;

cms.; (c) 10π cms.; (d)

52π
3

revs. per sec.; (b) 96π rads. per min.; (c)

π
;
20

(h)

31π
.
20

cms.
8π
5

rads. per sec.

metres.

5. 7.13 square cms. and 71.41 square cms.
6. sin θ = − 35 .
7. The chord has a length of 38.6cms. approximately.
8. The top of ladder is 11 metres from the ground and the length of the ladder is 11.7
metres.

y 1

−4π

−3π

−2π

−π

0

π

2π

3π

x✲

−1

The graph illustrates that
sin(θ + 2π) ≡ sin θ
and we say that sinθ is a “periodic function with period 2π”.
Other numbers which can act as a period are ±2nπ where n is any integer; but 2π itself is
the smallest positive period and, as such, is called the “primitive period” or sometimes
the “wavelength”.
We may also observe that
sin(−θ) ≡ − sin θ


and so cosθ, like sinθ, is a periodic function with primitive period 2π
We may also observe that
cos(−θ) ≡ cos θ
which makes cosθ what is called an “even function”.
3. y = tan θ
y ✻

−π

0

− π2

π
2

π

x

✲

This time, the graph illustrates that
tan(θ + π) ≡ tan θ
which implies that tanθ is a periodic function with primitive period π.
We may also observe that
tan(−θ) ≡ − tan θ
which makes tanθ an “odd function”.
3.2.2 GRAPHS OF MORE GENERAL TRIGONOMETRIC

2 2 2 2

(c) The y-axis must be placed between the smallest negative intersection with the
θ-axis and the smallest positive intersection with the θ - axis (in proportion to their
values). In this case, the y-axis must be placed half way between θ = − π2 and θ = π2 .
✻

y 5

−4π

−3π

−2π

−π

0

π

2π

3π

x✲

−5

Of course, in this example, from earlier trigonometry results, we could have noticed


1.07

2.64

4.21

x✲

−3

3.2.3 EXERCISES
1. Make a table of values of θ and y, with θ in the range from 0 to 2π in steps of
hence, sketch the graphs of
(a)
y = sec θ;
(b)
y = cosec θ;
(c)
y = cot θ.
2. Sketch the graphs of the following functions:
(a)
y = 2 sin θ +

π
;
4

(b)
y = 2 cos(3θ − 1).

O

π
2

3π
2

✲θ

−1

(b) The graph is

y
✻
1

−2π

−π

O

π

2π

✲θ


4

− 13π
4

− 9π
4

− 5π
4

− π4

3π
4

7π
4

11π
4

0.86

1.91

2.96

x✲



5

−2.09

−1.64 −1.19

−0.74

−0.29

0.16

0.61

1.06

5π
6

11π
6

x✲

−5

y

(d) The graph is

UNIT NUMBER
3.3
TRIGONOMETRY 3
(Approximations & inverse functions)
by
A.J.Hobson
3.3.1
3.3.2
3.3.3
3.3.4

Approximations for trigonometric functions
Inverse trigonometric functions
Exercises
Answers to exercises


UNIT 3.3 - TRIGONOMETRY
APPROXIMATIONS AND INVERSE FUNCTIONS
3.3.1 APPROXIMATIONS FOR TRIGONOMETRIC FUNCTIONS
Three standard approximations for the functions sin θ, cos θ and tan θ respectively can be
obtained from a set of results taken from the applications of Calculus. These are stated
without proof as follows:

sin θ = θ −

θ3 θ5 θ7
+
− .....
3!


tan θ

θ.

Better approximations are obtainable if more terms of the infinite series are used.
EXAMPLE
Approximate the function
5 + 2 cos θ − 7 sin θ
to a quartic polynomial in θ.
Solution
Using terms of the appropriate series up to and including the fourth power of θ, we deduce
that
θ4
θ3
− 7θ + 7
5 + 2 cos θ − 7 sin θ 5 + 2 − θ2 +
12
6
1 4
=
θ + 14θ3 − 12θ2 − 84θ + 84 .
12

1


3.3.2 INVERSE TRIGONOMETRIC FUNCTIONS
It is frequently necessary to determine possible angles for which the value of their sine, cosine
or tangent is already specified. This is carried out using inverse trigonometric functions

Tan ( 3) = 60 ± n180
THat is, angles in opposite quadrants have the same tangent.
Another Type of Question
3. Obtain all of the solutions to the equation
cos 3x = −0.432
which lie in the interval −180◦ ≤ x ≤ 180◦ .
Solution
This type of question is of a slightly different nature since we are asked for a specified
selection of values rather than the general solution of the equation.
2


We require that 3x be any one of the angles (within an interval −540◦ ≤ 3x ≤ 540◦ )
whose cosine is equal to −0.432. Using a calculator, the simplest angle which satisfies
this condition is 115.59◦ ; but the complete set is
±115.59◦ ± 244.41◦ ± 475.59◦
Thus, on dividing by 3, the possibilities for x are
±38.5◦ ± 81.5◦ ± 158.5◦
Note: The graphs of inverse trigonometric functions are discussed fully in Unit 10.6, but
we include them here for the sake of completeness
y = Sin−1 x

y = Cos−1 x

✻

✻
q 2π
qπ



✲

r − π2

Of all the possible values obtained for an inverse trigonometric function, one particular one
is called the “Principal Value”. It is the unique value which lies in a specified range
described below, the explanation of which is best dealt with in connection with differential
calculus.
To indicate such a principal value, we use the lower-case initial letter of each inverse function.
(a) θ = sin−1 x lies in the range − π2 ≤ θ ≤ π2 .
(b) θ = cos−1 x lies in the range 0 ≤ θ ≤ π.
3


(c) θ = tan−1 x lies in the range − π2 ≤ θ ≤ π2 .
EXAMPLES
1. Evaluate sin−1 ( 21 ).
Solution
sin−1 ( 21 ) = 30◦ or π6 .
√
2. Evaluate tan−1 (− 3).
Solution
√
tan−1 (− 3) = −60◦ or − π3 .
3. Write down a formula for u in terms of v in the case when
v = 5 cos(1 − 7u).
Solution
Dividing by 5 gives


1. If powers of θ higher then three can be neglected, find an approximation for the function
6 sin θ + 2 cos θ + 10 tan θ
in the form of a polynomial in θ.
2. If powers of θ higher than five can be neglected, find an approximation for the function
2 sin θ − θ cos θ
in the form of a polynomial in θ.
4


3. If powers of θ higher than two can be neglected, show that the function
θ sin θ
1 − cos θ
is approximately equal to 2.
4. Write down the principal values of the following:
(a) Sin−1 1;
(b) Sin−1 − 12 ;
(c) Cos−1 −

√

3
2

;

−1

(d) Tan 5;
√
(e) Tan−1 (− 3);

(a) v = sin u;
(b) v = cos 2u;
(c) v = tan(u + 1).
5


8. If x is positive, show diagramatically that
(a)
sin−1 x + cos−1 x =

π
;
2

(b)
√
sin−1 x = cos−1 1 − x2 .
3.3.4 ANSWERS TO EXERCISES
1.

7θ3
3

− θ2 + 16θ + 2.

2. θ +

θ3
6


7. (a) u = Sin−1 v; (b) u = 21 Cos−1 v; (c) u = Tan−1 v − 1.
8. A suitable diagram is

1 ✟✟
✟
✟✟

✟
✟✟
✟
✟

x

✟✟

✟
✟✟

√

1 − x2

6


“JUST THE MATHS”
UNIT NUMBER
3.4
TRIGONOMETRY 4

❅
✟✟
✟
❅
✟
✟
❅
b
a
✟
❅
✟✟
✟
❅
✟✟
❅
✟
✟
❅
✟
✟
❅
✟
✟
❅
✟
❅
A ✟✟
❅ B



15m

✁
✁
✁
✁
✁
✁
✁
✁θ

7m
From Pythagoras’ Theorem, the length of the cable will be
√
72 + 152 16.55m.
The angle of inclination to the horizontal will be θ, where
15
tanθ = .
7
Hence, θ

65◦ .

3.4.3 THE SINE AND COSINE RULES
Two powerful tools for the solution of triangles in general may be stated in relation to the
earlier diagram as follows:
(a) The Sine Rule
a
b