3. ELEMENTS
So originally trigonometry was understood to define relations between elements of a
triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line
segments may serve as the sides of a triangle. They do if they satisfy the triangle
inequality, or rather three triangle inequalities. Not any three angles may be the angles of
a triangle. In Euclidean geometry, the three angles of a triangle add up to a straight angle.
These requirements impose limitations on the manner in which the relations between the
elements are defined. In modern trigonometry these relations are extended to arbitrary
angles. This can be done, for example, by observing the projections of arotating radius of
a circle and a tangent at the end of the radius.
One of the oldest surviving fragments of Hipparcus for
trigonometry Elements, found from the Histories VIII
on Papyrus Oxyrhynchus 2099, early 2nd century AD
4. Ratios of trigonometry
1. Sine of <A=side opposite to < A BC
hypotenuse AC
2. Cos of <A=side adjacent to <A BA
hypotenuse AC
3. Tan of <A= side opposite to <A BC
side adjacent to <A BA
4. Cosec of <A= 1 AC
sine of <A BC
5. Sec of <A= 1 AC
cosine of <A AB
6. Cot of <A= 1 AB
tangent of <A BC C B
A
5. Trigonometric ratios of some special
angles
1. Trigonometric ratio of 45˚is:
sin 45˚=1/√2 cosec 45˚=√2
cos 45˚=1/√2 sec 45˚=√2
tan 45˚=1 cot 45˚=1
6. 2. Trigonomteric ratios of 30˚ and 60˚ are:
sin 30˚=1/2 sin 60˚=√3/2
cos 30˚=√3/2 cos 60˚=1/2
tan 30˚=1/√3 tan 60˚=√3
cosec 30˚=2 cosec 60˚=2/√3
sec 30˚=2/√3 sec 60˚=2
cot 30˚=√3 cot 60˚=1/√3
7. 3. Trigonometric ratios of 0˚ and 90˚ are:
sin 0˚=0 sin 90˚=1
cos O˚=! cos 90˚=0
tan 0˚=0 tan 90˚=!
cosec 0˚=! cosec 90˚=1
sec 0˚=1 sec 90˚=!
cot 0˚=! cot 90˚=0
8. Trigonometric ratios of
complementary angles
sin(90˚-A)= cos A
tan(90˚-A)=cot A
sec(90˚-A)=cosec A
cos(90˚-A)=sin A
cot(90˚-A)=tan A
cosec(90˚-A)=sec A
for all the values
of angle A lying
between 0 ˚and
90˚. Check
whether this
holds for A=˚ or
A=90˚
10. Trigonometry’s
Achievements
Trigonometryis not only used in mathematics but it is also used in life aroundus.
Trigonometryis oneofthe most ancient subjects studied by the scholarsall over
the world. Trigonometrywas invented for it’s need in astronomy.Since then
astronomerhave usedit, for instance, tocalculate distances from the Earthto the
planets and stars.Trigonometry hasalso spread it’s reachto geography too. It also
took navigation along with it. Theknowledge oftrigonometry is usedto construct
maps, determine the position ofan island in relation to the longitudes and
latitudes.
