Top Careers & You
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Trigonometry
Angles and their relationship
Angles are measured in many units viz. degrees, minutes, seconds, radians, gradients.
Where 1 degree = 60 minutes, 1 minute = 60 seconds,
radians = 180 = 200
1 radian = 180/ and 1 degree = / 180 radians.
The angle at the centre is of 1 radian
rr
Do you know?
1 radian is the angle
made at the centre by
the
arc
of
length
equal to the radius of
the circle.
Basic Trigonometric Ratios
In a right triangle ABC, if be the angle between AC & BC.
A
If is one of the angle other then right angle, then the side opposite to the angle is perpendicular (P) and the
sides containing the angle are taken as Base ( B) and the hypotenuse (H). In this type of triangles, we can
have six types of ratios. These ratios are called trigonometric ratios.
Sin =
P
B
, Cos =
H
H
Cosec =
H
,
P
Tan =
Sec =
P
B
H
B
, Cot =
B
P
Important Formulae
For any angle :
1.
sin2 + cos2 = 1
2.
1 + tan2 = sec2
3.
1 + cot = cosec
[Note sin2 = (sin )2 and not (sin 2)]
2
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Range of Values of Ratios
If 0 360o, then the values for different trigonometric ratios will be as follows.
1.
1 Sin 1
2.
1 cos 1
Maximum and
3.
tan
Minimum values of sin
4.
cot
5.
Sec -1
&
1 Sec
6.
Cosec 1
&
1 Cosec
TIP
or cos are + 1 and
1 respectively.
Sign of Trigonometric ratios
We divide the angle at a point (i.e. 360) into 4 parts called quadrants. In the first quadrant all the
trigonometric ratios are positive
II
90 180
180 270
0 90
270 360
III
IV
SOME MORE RESULTS:
TIP
T. Ratios
Angles
Sin
Cos
Tan
Cot
Sec
Cosec
90 -
Cos
Sin
Cot
Tan
Cosec
Sec
Sugar: Sin positive
90 +
Cos
Sin
Cot
Tan
Cosec
Sec
To: Tan positive
180 -
Sin
Cos
Tan
Cot
Sec
Cosec
180 +
Sin
Cos
Tan
Cot
Sec
Cosec
270 -
Cos
Sin
Cot
Tan
Cosec
Sec
270 +
Cos
Sin
Cot
Tan
Cosec
Sec
360 -
Sin
Cos
Tan
Cot
Sec
Cosec
360 +
Sin
Cos
Tan
Cot
Sec
Cosec
Add: All positive
Ex.1 Simplify
Sol.
Coffee : Cos positive
tan( 90 o + ) sin( 180 o + ) sec( 270 o + )
.
cos( 270 o ) cos ec (180 o ) cot( 360 o )
tan (90o + ) = cot ,
sin ( 180o + ) = sin
sec (270o + ) = cosec ,
cos (270o ) = sin
cosec (180o ) = cosec ,
cot (360o ) = cot
Given expression =
( cot )( sin )(cos ec )
= 1.
( sin )(cos ec )( cot )
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Ex.2 If cot A =
Sol.
cot A =
3
4
tan A =
3
, find the value of 3 cos A + 4 sin A, where A is in the first quadrant.
4
4
3
Perpendicular 4
=
base
3
Draw a right PQR in which Q = A,
PR = 4, QR = 3
PQ = (PR ) 2 + (QR ) 2
=
( 4 ) 2 + (3 ) 2 =
16 + 9 =
25
PQ = 5 units
cos A =
cos A =
base
Hypotenuse
PR 4
3
=
and sin A =
5
PQ 5
3 cos A + 4 sin A = 3
3 cos A + 4 sin A =
3
4
9 16
+ 4 = +
5
5
5
5
25
= 5.
5
sin 35 o
Ex.3 Find the value of
o
cos 55
Sol.
sin 35 o
cos 55 o
+ cos 55
o
sin 35
+ cos 55
sin 35 o
2 cos 60 o
2 cos 60 o
sin(90o 55o )
cos(90o 35o )
1
=
+
2
o
sin 35
2
cos 55
o
cos 55o
+ sin 35 1
=
cos 55o
sin 35o
Q sin (90 o ) = cos
cos ( 90 o ) = sin
=1+11=1
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Values of trigonometric Ratio for some special angles:
30
45
60
90
1
sin
1/2
1/2
3/2
cos
3/2
1/2
1/2
tan
1/3
(Not defined)
Properties of triangle
Sine Rule
In any triangle ABC if AB, BC, AC be represented by c, a, b respectively
a
b
c
=
=
= 2R
then we have
sin A
sin B
sin C
c
Where R is circum radius =
b
R
abc
4 Area of triangle
Cosine Rule
In a triangle ABC of having sides of any size, we have the following rule;
Cos A =
b2 + c 2 a2
2bc
Cos B =
c 2 + a2 b2
2ac
Cos C =
a2 + b2 c 2
2ab
Area of triangle
Area =
=
1
1
1
bc sin A = ac sin B = ab sin C
2
2
2
S (S a ) (S b ) (S c )
where S = semi perimeter
Ex.4 Solve the equation sec2x 2 tan x = 0
Sol. sec2x 2 tan x = 0
1 + tan2x 2 tan x = 0
tan2x 2 tan x + 1 = 0
(tan x 1)2 = 0
tan x = 1
x = 45o.
{ Q sec2 tan2 = 1}
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