Download A Brief Introduction to Sine and Cosine

A Brief Introduction to Sine and Cosine
Trigonometry may be viewed, in the first instance, as the study of triangles. The essential ingredients come from plane Euclidean geometry.
(i) The initial definition of sine and cosine depends on the ratios of the lengths of corresponding sides of similar triangles being the same.
(ii) The relationship between the sine and the cosine of an acute angle is given by Pythagoras’
Theorem.
(iii) The definition of radian measure of angles depends on the ratio between the arc lengths
of circular arcs subtending a fixed angle at the centre of their circles and the radii of these
circles being constant.
1. The Definition of Sine and Cosine for Acute Angles
Two triangles are similar, if and only if they have the same three angles.
If one of the angles is a q and another is θ, then we may depict two similar, but not congruent
triangles, as follows.
θ
Since the triangles are similar, the ratios of the lengths of corresponding sides are equal. We label
the sides by their respective lengths
H
B
h
b
θ
a
A
and obtain
a
b
h
=
= ,
A
B
H
form which we deduce that
b
B
=
h
H
and
a
A
=
h
H
Since these two ratios depend only on the angle θ in a right-angled triangle, and not on the
particular triangle drawn they define functions of θ.
1
2
Definition. The sine of the acute angle θ, written sin θ, is the ratio of the length of the opposite
side to the length of the hypotenuse of any right-angled triangle which has θ as one of its angles.
The cosine of θ, written cos θ, is the ratio of the length of the adjacent side to the length of the
hypotenuse.
In the above diagrams
b
and
sin θ :=
h
so that
h sin θ = b
and
cos θ :=
a
h
h cos θ = a
2. The Relationship Between Sine and Cosine
Since a triangle defining the value of the acute angle, θ, is right-angled, Pythagoras’ Theorem
asserts that a2 + b2 = h2 , or, dividing by h2 ,
a2
b2
+
= 1.
h2
h2
This is equivalent to
cos2 θ + sin2 θ = 1.
3. Radian Measure and Sine and Cosine for Arbitrary Real Numbers
Measuring angles in degrees is a tradition going back to Babylonian times. The full circle is
divided into 360 degrees because the number 6 and its multiples had religious/mystical/magical
significance. This arbitrary definition is mathematically inconvenient.
An alternative measure of angle, devoid of non-mathematical connotations, is best suited to mathematics. It is based on the observation, known in Ancient Greece, that for any circle, the ratio of
the circumference to the diameter is constant. This number is π.
Consequently, if we take an arc of a circle which subtends the angle θ at the centre of the circle,
then the ratio of the length of that arc and the radius of the circle — half the length of a diameter
of that circle — is independent of the circle itself.
L
l
This means that =
in the diagram
r
R
r
l
L
θ
R
Hence we may take this ratio as a measure of angle. Being the ratio of two lengths, it is a pure
l
number. We define the radian measure of the angle θ to be in the diagram above, so that,
r
rθ = l
Consequently, if we take a unit circle — a circle with radius one unit of length — then r = 1 and
so the radian measure of the angle subtended by an arc is then the numerical value of the length
of that arc. Notice that this is independent of the particular units of length chosen.
Since the circumference of the unit circle is 2π units of length, we see that 360 degrees corresponds
to 2π radians.
3
Even more, this allows us to regard any real number as the measure of some angle, and extend
the definition of sine and cosine to all real numbers.
4. The Sine and Cosine of an Arbitrary Real Number
As we may use a circle of any radius when determining the sine and cosine of an acute angle, we
may use a unit circle in the Cartesian plane, which is
{(x, y) ∈ R2 | x2 + y 2 = 1}
We inscribed our triangle by taking the X-axis as the “base”, and drawing a ray from the origin,
(0, 0) making an angle of θ in the anti-clockwise direction. Let its intersection with the unit
circle have co-ordinates (x, y). Then the perpendicular from this point meets the X-axis at (x, 0),
completing our triangle.
Y
(0, 1)
(x, y)
1
θ
(0, 0)
(1, 0)
(x, 0)
X
Since the hypotenuse of our triangle has length 1, we see that the side opposite the angle a the
origin has length sin θ and the horizontal side has length cos θ. Moreover, the length of the path
from the point (1, 0) to (x, y) clockwise along the unit circle is θ.
(x, y)
1
sin θ
θ
θ
cos θ
π
It is immediate that for the acute angle θ — that is, 0 < θ , for angles measured in radians —
2
x = cos θ
and
y = sin θ
This allows us to extend the definition of sin θ and cos θ to any real number θ, using radian measure
for angles.
We travel |θ| units along the unit circle starting from (1, 0), clockwise if θ < 0 and anti-clockwise
if θ > 0, reaching a point with co-ordinates (x, y). We define
cos θ := x
and
sin θ := y.
Since (x, y) lies on the unit circle, it is immediate that
cos2 θ + sin2 θ = 1.
4
5. Properties of Sine and Cosine
Given real numbers A, B,
(1) sin(A ± π) = − sin A
cos(A ± π) = − cos A
(2) sin(−A) = −(sin A)
cos(−A) = cos A
(3) sin 0 = 1
cos 0 = 1
(4) sin π2 = 1
cos π2 = 0
(5) sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
The first four of these can be proved directly from the diagram above defining sine and cosine for
arbitrary real numbers.
As to the last, let 4OAB be a right-angled triangle, with the angle at O being β. Let the length
of the hypotenuse, OA, be one unit. Let OB be inclined at an angle α from the horizontal.
A
1
B
β
α
O
The length of the side OB is cos β, and the length of the side AB is sin β.
Construct a vertical edge through A to form a triangle OAC with a right-angel at C.
A
1
β
α
O
C
Then the side OC has length cos(α + β) and the side AC has length sin(α + β)).
Let the perpendicular through B meet the line through OC at D, and let the perpendicular
through B meet AC at E. We obtain a right-angled triangle 4OBD, with angle α at O, a
right-angled triangle 4SAB, with angle α at B, and a rectangle ECDB.
5
A
α
1
E
α
B
β
α
O
C
D
The length of AC is the sum of the lengths of AE and EC.
The right-angled triangle, 4AEB, has hypotenuse AB of length sin β and angle α at A. Hence,
the length of AE is
cos α sin β
From the rectangle ECDB, we see that the length of EC is the same as the length of BD.
The right-angled triangle, OBD has hypotenuse OB of length cos β and angle α at O. Thus, the
length of BD, and so also of EC, is
sin α cos β.
Since the length of AC is sin(α + β), we obtain
sin(α + β) = sin α cos β + cos α sin β.
The proof that
cos(α + β) = cos α cos β − sin α sin β
follows by a similar argument using the same diagrams, and is left to the reader as an exercise.