1. INTRODUCTION
TO
TRIGONOMETRY
Submitted By
AMAL A S

2. Trigonometry...?????
The word trigonometry is derived from the
ancient Greek language and means
measurement of triangles.
trigonon “triangle”
+
metron “measure”
=
Trigonometry

3. Trigonometry ...
is all about
Triangles…
C
a
b
B
A
c

4. A right-angled triangle (the right angle is
shown by the little box in the corner) has
names for each side:
Adjacent is adjacent to the angle "θ“
Opposite is opposite the angle
 The longest side is the Hypotenuse.
Opposite
Right Angled Triangle
θ
Adjacent

5. DEGREE MEASURE AND RADIAN MEASURE
B
B
1
1
1
A
O
1
A
O
Initial Side
Degree measure: If a rotation from the
initial side to terminal side is(1/360)th of
a revolution, the angle is said to have a
measure of one degree, written as 1 .
Radian measure: Angle
subtended at the centre by an arc
of length 1 unit in a unit circle
(circle of radius 1 unit) is said to
have measure of 1 radian.
Degree measure= 180/ π x Radian
measure
Radian measure= π/180 x Degree
measure

6. ANGLES
Angles (such as the angle "θ" ) can be
in Degrees or Radians.
Here are some examples:
Angle
Degree
Radians
Right Angle
90°
π/2
Straight Angle
180°
π
Full Rotation
360°
2π

7. Trigonometric functions..

8. "Sine, Cosine and Tangent"
The three most common functions in trigonometry are
Sine, Cosine and Tangent.
They are simply one side of a triangle divided by another.
Sine Function:
sin(θ) = Opposite / Hypotenuse
Cosine Function:
cos(θ) = Adjacent / Hypotenuse
Tangent Function:
tan(θ) = Opposite / Adjacent
Opposite
For any angle "θ":
θ
Adjacent

9. Their graphs as functions and the
characteristics
•

10. TRGONOMETRIC FUNCTIONS
π/6
π/4
π/3
π/2
π
3π/2
2π
30°
0°
45°
60°
90°
180°
270°
360°
sin
0
1/2
1/√2
√3/2
1
0
-1
0
cos
1
√3/2
1/√2
1/2
0
-1
0
1
tan
0
1/√3
1
√3
Not
defined
0
Not
defined
0

11. Other Functions (Cotangent, Secant, Cosecant)
Cosecant Function : csc(θ) = Hypotenuse / Opposite
Secant Function : sec(θ) = Hypotenuse / Adjacent
Cotangent Function : cot(θ) = Adjacent / Opposite
Opposite
 Similar to Sine, Cosine and Tangent, there are three other trigonometric
functions which are made by dividing one side by another:
θ
Adjacent

12. Proof for trigonometric ratios
30 ,45 ,60

13. Computing unknown sides or angles in a right triangle.
 In order to find a side of a right triangle you can use the Pythagorean
Theorem, which is a2+b2=c2. The a and b represent the two
shorter sides and the c represents the longest side which is the
hypotenuse.
To get the angle of a right angle you can use sine, cosine, and
tangent inverse. They are expressed as tan^(-1) ,cos^(-1) , and sin^(1) .

14. o Find the sine, the cosine, and the tangent of 30 .
Begin by sketching a 30 -60 -90 triangle. To
make the calculations simple, you can choose 1 as
the length of the shorter leg. From Pythagoras
Theorem , it follows that the length of the longer
leg is √3 and the length of the hypotenuse is 2.
sin 30 = opp./hyp. = 1/2 = 0.5
cos 30 = adj./hyp. = √3/2 ≈ 0.8660
2
1
30
tan 30 = opp./adj. = 1/√3 = √3/3 ≈ 0.5774
√3

15. o Find the sine, the cosine, and the tangent of 45 .
Begin by sketching a 45 -45 -90 triangle.
Because all such triangles are similar, you
can make calculations simple by choosing 1
as the length of each leg. The length of the
hypotenuse is √2 (Pythagoras Theorem).
sin 45 = opp./hyp. = 1/√2 =2/√2≈ 0.7071
cos 45 = adj./hyp. = 1/√2 =2/√2≈ 0.7071
√2
1
45
tan 45 = opp./adj. = 1/1 = 1
1

16. o Find the sine, the cosine, and the tangent of 60 .
Begin by sketching a 30 -60 -90 triangle. To
make the calculations simple, you can choose 1
as the length of the shorter leg. From
Pythagoras Theorem , it follows that the length
of the longer leg is √3 and the length of the
hypotenuse is 2.
sin 60 = opp./hyp = √3/2 ≈ 0.8660
60
2
1
cos 60 = adj./hyp = ½ = 0.5
tan 60 = opp./adj. = √3/1 ≈ 1.7320
30
√3

17. TRIGONOMETRIC IDENTITIES

18.  Reciprocal Identities
 Pythagorean Identities
sin u = 1/csc u
sin2 u + cos2 u = 1
cos u = 1/sec u
1 + tan2 u = sec2 u
tan u = 1/cot u
1 + cot2 u = csc2 u
csc u = 1/sin u
 Quotient Identities
sec u = 1/cos u
tan u = sin u /cos u
cot u = 1/tan u
cot u =cos u /sin u

19.  Co-Function Identities
 Parity Identities (Even & Odd)
sin( π/2− u) = cos u
sin(−u) = −sin u
cos( π/2− u) = sin u
cos(−u) = cos u
tan( π/2− u) = cot u
tan(−u) = −tan u
cot( π/2− u) = tan u
cot(−u) = −cot u
csc( π/2− u) = sec u
csc(−u) = −csc u
sec( π/2− u) = csc u
sec(−u) = sec u

20.  Sum & Difference Formulas
sin(u ± v) = sin u cos v ± cos u sin v
cos(u ± v) = cos u cos v ∓ sin u sin v
tan(u ± v) = tan u ± tan v / 1 ∓ tan u tan v
 Double Angle Formulas
sin(2u) = 2sin u cos u
cos(2u) = cos2 u − sin2 u
= 2cos2 u − 1
= 1 − 2sin2 u
tan(2u) =2tanu /(1 − tan2 u)

21.  Sum-to-Product Formulas
Sin u + sin v = 2sin [ (u + v) /2 ] cos [ (u − v ) /2 ]
Sin u − sin v = 2cos [ (u + v) /2 ] sin [ (u − v ) /2 ]
Cos u + cos v = 2cos [ (u + v) /2 ] cos [ (u − v) /2 ]
Cos u − cos v = −2sin [ (u + v) /2 ] sin [ (u − v) /2 ]

22.  Product-to-Sum Formulas
Sin u sin v = ½ [cos(u − v) − cos(u + v)]
Cos u cos v = ½ [cos(u − v) + cos(u + v)]
Sin u cos v = ½ [sin(u + v) + sin(u − v)]
Cos u sin v = ½ [sin(u + v) − sin(u − v)]

23.  Power – Reducing / Half Angle Formulas
sin2 u = 1 − cos(2u) / 2
cos2 u = 1 + cos(2u) / 2
tan2 u = 1 − cos(2u) / 1 + cos(2u)