Angle and Its Measurement
Angle:
an angle may be defined as a measure of the rotation of a half ray about its origin. An angle XOP is positive if it is traced by a ray revolving in the anticlockwise direction and negative if
it is traced by a ray revolving in the clockwise direction.
Measurement of an Angle:
There are three systems of measurement of an angle:
i) Sexagesimal System or English System
ii) Centesimal system or French System
iii) Circular System
Sexagesimal System (degree measure):
In this system an angle is measured in degrees, minutes and seconds. One complete revolution is 360^{0} where one degree is written as 1^{0}. Further. Thus 1^{0} = 60 minutes (or 60') and 1'
= 60 seconds (or 60''). An angle of 90^{0} is also called a right angle.
Centesimal System (grade measure):
In this system an angle is measured in grades, minutes and seconds.
Here 1 right angle = 100 grades, written as 100^{g},
1 grade = 100 minutes, written as 100',
and 1 minute = 100 seconds, written as 100''.
Circular System (radian measure):
In this system an angle is measured in radians. A radian is an angle subtended at the centre of a circle by an arc whose length is equal to its radius.
AOB = 1 radian (written as 1^{C}).
Since the whole circle subtends an angle of 360^{0} (4 right angles) at the centre and the angles at the centre of a circle are in the ratio of substending arcs,
so that
AOB =
1 radian = × 4 right angles = × right angle
p radians = 2 right angles = 180^{0} = 200 g
1 radian = degree = = 57^{0}16'22'' nearly.
Trigonometric Ratios or Circular Functions:
In the above figures, Let OM = x, MP = y and OP = r > 0. The circular functions are defined as:
(i). = sin
(ii). = cos
(iii). = tan , x 0
(iv). = cot , y 0
(v). = cosec , y 0
(vi). = sec , x 0
Trigonometric ratios (or functions) may also be defined with respect to a triangle.
In a right angled triangle ABC, CAB = A and BCA = 90^{o} = /2. With reference to angle A, the six trigonometric ratios are:
is called the sine of A, and written as sinA.
is called the cosine of A, and written as cosA.
is called the tangnet of A, and written as tanA.
Obviously,. The reciprocals of sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. We write these as cosecA, secA, cotA
respectively. Since the hypotenuse is the greatest side in a right angle triangle, sinA and cosA can never be greater than unity and cosecA and secA can never be less than unity. Hence
sin A 1, cos A 1,cosec A 1, sec A1, while tan A and cot A may take any numerical value.
Note:
All the six trigonometric functions have got a very important property in common that is of periodicity.
Remember that the trigonometrical ratios are real numbers and remain same as long as angle A is real.
Signs of Trigonometric Ratios:
The following table describes the signs of various trigonometric ratios:
Ist quadrant  IInd quadrant  IIIrd quadrant  IVth quadrant 
All positive  sin , cosec positive  tan , cot positive  sec , cos positive 
Trigonometrical Identities:
An identity is a relation which is true for all values of the independent variable. There are three fundamental identities involving trigonometrical ratios:
i) sin^{2} + cos^{2} = 1
ii) 1 + tan^{2} = sec^{2}
iii) 1 + cot^{2} = cosec^{2}
Trigonometric Ratios of Standard Angles
Angle Ratios  Sin  Cos  Tan  Cosec  Sec  Cot 
0^{o}  0  1  0  Not defined  1  Not defined 
30^{o}  2  
45^{o}  1  1  
60^{o}  2  
90^{o}  1  0  Not defined  1  Not defined  0 
Allied Angles
Table  I
equals  sin  cos  tan  cot  sec  cosec 
   sin  cos  tan   cot  sec  cosec 
90^{o}   cos  sin  cot  tan  cosec  sec 
90^{o} +  cos   sin  cot   tan  cosec  sec 
180^{o}  sin   cos   tan   cot   sec  cosec 
180^{o}+   sin   cos  tan  cot   sec  cosec 
360^{o}   sin  cos   tan   cot  sec  cosec 
360^{o}+  sin  cos  tan  cot  sec  cosec 
Note:
Angle
and 90
^{o}

are complementary angles,
and 180
^{o}
 q are supplementary angles
sin(np + ( 1)
^{n}
) = sin
, n
I
cos(2np
) = cos
, n
I
tan(np +
) = tan
, n
I
i.e. sine of general angle of the form np + (1)
^{n}
will have same sign as that of sine of angle
and so on. The same is true for the respective reciprocal functions also.
Trigonometric Ratios of Compound Angles:
The Addition Formulae:
(i). sin (A + B) = sin A cos B + cos A sin B
(ii). cos (A + B) = cos A cos B  sin A sin B
(iii). tan (A + B) =
Subtraction Formulae:
(i). sin (A  B) = sin A cos B  cos A sin B
(ii). cos (A  B) = cos A cos B + sin A sin B
(iii). tan (A  B) =
Some Important Deductions:
(i). sin (A + B) sin (A  B) = sin
^{2}
A  sin
^{2}
B
(ii). cos (A + B) cos (A  B) = cos
^{2}
A  sin
^{2}
B
(iii). cot (A + B) =
(iv). cot (A  B) =
Transformation Formulae:
(a) Transformation of products into sums or differences:
(i). 2 sin A cos B = sin (A + B) + sin (A  B)
(ii). 2 cos A sin B = sin (A + B)  sin (A  B)
(iii). 2 cos A cos B = cos (A + B) + cos (A  B)
(iv). 2 sin A sin B = cos (A  B)  cos (A + B)
The above formulae can be easily derived by taking the sum and the difference of the addition and subtraction formulae.
(b) Transformation of sums or differences into products:
(i). sin (A + B) + sin (A  B) = 2 sin A cos B
or, sin C + sin D = 2 sin
cos
(ii). sin (A + B)  sin (A  B) = 2 cos A sin B
or, sin C  sin D = 2 cos
sin
(iii). cos (A + B) + cos (A  B) = 2 cos A cos B
or, cos C + cos D = 2 cos
cos
(iv). cos (A  B)  cos (A + B) = 2 sin A sin B
or, cos C  cos D = 2 sin
sin
Here A  B = C and A + B = D
A =
and B =
.
(v). tan A + tan B =
.
Trigonometric Ratios of Multiple Angles:
(i). sin 2A = 2 sin A cos A
(ii). cos 2A = cos
^{2}
A  sin
^{2}
A = 2 cos
^{2}
A  1 = 1  2 sin
^{2}
A
(iii). tan 2A =
(iv). sin 3A = 3 sin A  4 sin
^{3}
A = 4 sin (60
^{0}
 A) sin A sin (60
^{0}
+ A)
(v). cos 3A = 4 cos
^{3}
A  3 cos A = 4 cos (60
^{0}
 A) cos A cos (60
^{0}
+ A)
(vi). tan 3A =
= tan (60
^{0}
 A) tan A tan (60
^{0}
 A).
Trigonometric Ratios of Submultiple of an Angle:
(i). sin A = 2 sin
cos
(ii). cos A = cos
^{2}
 sin
^{2}
= 2 cos
^{2}
 1 = 1  2 sin
^{2}
(iii). tan A =
(iv). sin A = 3 sin
 4 sin
^{3}
(v). cos A = 4 cos
^{3}
 3 cos
(vi). tan A =
or sin
+cos
=
or sin
 cos
=
tan
=
The ambiguities of signs are removed by locating the quadrant in which
lies.
 a cosA + b sinA 
Also cosA
sinA =
Notes:
Any formula that gives the value of sin
in terms of sinA shall also give the value of sine of
.
Any formula that gives the value of cos
in terms of cosA shall also give the value of cos of
.
Any formula that gives the value of tan
in terms of tanA shall also give the value of tan of
(a) For any angles A, B, C:
sin (A + B +C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC  sinA sinB sinC
cos (A + B +C) = cosA cosB cosC cosA sinB sinC  sinA cosB sinC  sinA sinB cosC
;
(b) If A, B, C are the angles of a triangle ( or A + B + C =
), then
sinA cosB cosC + cosA sinB cosC + cosA cosB sinC = sinA sinB sinC
cosA sinB sinC + sinA cosB sinC + sinA sinB cosC = 1 + cosA cosB cosC
tanA + tanB + tanC = tanA tanB tanC
cotB cotC + cotC cotA + cotA cotB = 1
sin2A + sin2B + sin2C = 4sinA sinB sinC
cos2A + cos2B + cos2C = 14cosA cosB cosC
cos
^{2}
A + cos
^{2}
B + cos
^{2}
C = 1  2cosA cosB cosC
Graphs of Trigonometric Functions:
We have seen that all trigonometric function are periodic. Since sin (2
+ x) = sin x, cos (2
+ x) = cos x, and tan (
+ x) = tan x, the period of sine and cosine functions is 2p where as
the period of tangent function is
. Moreover, period of sin ax or cos ax is
and that of tan ax is
.