The chapter on trigonometric functions and equations typically covers the study of trigonometric functions, which are functions of an angle. These functions include sine, cosine, tangent, secant, cosecant, and cotangent. Students learn about the properties and graphs of these functions, as well as ...
Measure of Angles IN THIS CHAPTER ....
The measure of angle is the amount of rotation from the direction of one ray of Measure of Angles
the angle to the other. The initial and final positions of the revolving ray are Trigonometric Ratios
respectively called the initial and terminal sides (arms).
Trigonometric Functions
Q
Graph of Trigonometric Functions
Trigonometric Identities
Complementary and
Supplementary Angles
O P Variation of Values of
Trigonometric Ratios in Different
If the rotation is in clockwise sense, the angle measured is negative and if the
Quadrants
rotation is in anti-clockwise sense, the angle measured is positive.
Trigonometrical Ratios of Some
Useful Angles between 0° and 90°
System of Measurement of Angles
Trigonometric Ratios of
There are two systems of measurement
Compound Angles
Sexagesimal System Maximum and Minimum
In this system each angle is divided into 90 equal parts and each part is Greatest and Less Value of
known as a degree. Thus, a right angle is equal to 90 degrees. One degree is a sin q + b cos q
denoted as 1°. Trigonometric Equations
Each degree is divided into 60 equal parts each of which is known as one Solution of Trigonometric
minute. One minute is denoted as 1¢. Each minute is consist of 60 parts, each Equation of the Form
part is known as a second. One second is denoted by 1¢ ¢. a cos q + b sin q = c
Hence, 1 right angle = 90° ( 90 degree)
1 degree = 1° = 60 ¢ ( 60 minute)
1 minute = 1 ¢ = 60 ¢ ¢ (60 second)
,Trigonometric Function and Equations 193
Circular System Let ÐAOP = q
If the angle subtended by an arc of é arc AP q c lù
length l to the centre of circle of r
l êQ ÐAOP = radius OP = 1 = q , using q = r ú
ë û
l
radius r is q, then q = . q
Now, the six trigonometric functions may be defined as
r r
under
If the length of arc is equal to the
radius of the circle, then the angle OM PM
(i) cos q = =x (ii) sin q = =y
subtended at the centre of the circle OP OP
will be one radian. One radian is denoted by 1c. OP 1 OP 1
The ratio of the circumference of the circle to the (iii) sec q = = ,x ¹ 0 (iv) cosec q = = ,y¹ 0
OM x PM y
diameter of the circle is denoted by a greek letter p and it
is a constant quantity. PM y OM x
(v) tan q = = ,x ¹ 0 (vi) cot q = = ,y¹0
Circumference of circle OM x PM y
\ =p
Diameter of circle
Graph of Trigonometric Functions
Relation between Degree and Radian
The number of radians of an angle subtended by an arc of Graph of sin x
a circle at the centre is equal to the ratios of arc and Y
radius.
180° = p c
(– 3p2 , 1) ( p2, 1)
and 1 radian = 57°17 ¢ 44 . 8¢¢ y=1
B D
If measure of an angle is given in degree, then to convert h1 h1
p X¢ O
X
it into radians, multiply the measure of an angle by A p C
180°
–2p – 3p –p –p p 3p
2p
2 2 2 2
and if the measure of an angle is given in radians, then y=–1
to convert it into degree, write 180° at the place of p . ( – p , –1
2 ) ( 3p , –1
2 )
Y¢
Example 1. Given that the side length of rhombus the
geometric mean of the lengths of its diagonals. The radian Facts Related to sin x
measure of the acute angle of the rhombus is (a) Domain = R
p p p p
(a) (b) (c) (d) (b) Range = [-1, 1]
12 6 4 3
(c) Period = 2p
Sol. (b) Let side of rhombus is x units and D C (d) Graph of sin x is continuous for all real values of x.
diagonal of rhombus are a and b units.
b p/2
Given, x2 = ab Graph of cos x
1 2 ab a Y
Area of rhombus = x sin q = q
2 2 A B
x
1 (– 2p, 1) (0, 1) (2p, 1)
Þ sin q = = sin 30 º y=1
2 D
Þ q = 30 º O
X¢ X
p –2p – 3p –p –p
p p 3p 2p
In radian 180° ´ p = radius. 2 2 2 2
6 y = –1
(– p, –1) (p, –1)
Trigonometric Functions Y¢
Let X ¢ OX and YOY ¢ be the Y
Facts Related to cos x
coordinate axes. Taking O B
P(x, y) (a) Domain = R
as the centre and a unit q
1 y (b) Range = [-1, 1]
radius, draw a circle, q A
X¢
cutting the coordinate A¢ O x M
X (c) Period = 2p
axes at A, B, A¢ and B¢ , (d) Graph of cos x is continuous for all real values of x
as shown in the figure.
B¢
Y¢
,194 JEE Main Mathematics
Graph of tan x Facts Related to sec x
p
Y (a) Domain = R - ( 2n + 1) ,n ÎI
2
(b) Range = ( - ¥ , - 1] È [1, ¥ )
1
(c) Period = 2p
mp
X¢ –p p X (d) Graph of sec x is discontinuous at x = , where m
– 3p –p –p O p p 3p 2
2 2 4 4 2 2
–1 is an odd integer.
Graph of cosec x
Y¢
Y
Facts Related to tan x
p
(a) Domain = R - ( 2n + 1) ,n ÎI y = cosec x
2
(b) Range = ( - ¥ , ¥ )
(p2 ,1)
(c) Period = p
y=1
mp 1
(d) Graph of tan x is discontinuous at x = , where m y = sin x
2 X¢ –2p –p 3p X
–p/2 O p p 2p
– 3p
is an odd integer. 2 2 2
–1 y = –1
Graph of cot x 3p ,_1
(–2p,–1) (
2 )
Y
Y¢
X¢ X Facts Related to cosec x
–2p – 3p –p –p O p p 3p 2p
asymptotes
2 2 2 2
(a) Domain = R - np, n Î I
(b) Range = ( -¥ , - 1] È [1, ¥ )
(c) Period = 2p
Y¢
(d) Graph of cosec x is discontinuous at x = mp, where
Facts Related to cot x m is an integer.
(a) Domain = R - np, n Î I
(b) Range = ( - ¥ , ¥ ) Trigonometric Identities
(c) Period = p A trigonometric equation is an identity, if it is true for all
(d) Graph of cot x is discontinuous at x = mp, where m is values of the angle or angles involved.
an integer. Some Important Identities are Given Below
Graph of sec x (i) cos2 q + sin2 q = 1
Y or cos2 q = 1 - sin2 q
y = sec x
(–2p, 1) (2p, 1) or sin2 q = 1 - cos2 q
(ii) sec2 q - tan2 q = 1
y=1 or sec2 q = 1 + tan2 q
1
X¢
–2p – 3p –p O p p 3p 2p
X
y = cos x or tan2 q = sec2 q - 1
–p
2 2 2 2
–1 O y = –1 (iii) cosec2 q - cot2 q = 1
or cosec2 q = 1 + cot2 q
(–p, –1) Y¢ (p, –1) or cot2 q = cosec2 q - 1
, Trigonometric Function and Equations 195
Transformation of One Trigonometric Ratio to Another Trigonometric Ratios
sin q cos q tan q cot q sec q cosec q
tan q 1 2
(sec q - 1) 1
sin q sinq (1 - cos 2 q)
(1 + tan2 q) (1 + cot 2 q) sec q cosec q
1 cot q 1 (cosec 2 q - 1)
cos q 2
(1 - sin q) cos q
(1 + tan2 q) (1 + cot 2 q) sec q cosec q
sin q (1 - cos 2 q) 1 1
tan q tanq
2
(sec q - 1)
(1 - sin2 q) cos q cot q (cosec 2 q - 1)
2 cos q 1 1
(1 - sin q)
cot q cot q (cosec 2 q - 1)
sin q (1 - cos 2 q) tanq (sec 2 q - 1)
1 1 cosec q
(1 + cot 2 q)
sec q (1 + tan2 q) sec q
2
(1 - sin q) cos q cot q (cosec 2 q - 1)
1 1 2 sec q
(1 + tan q)
cosec q (1 + cot 2 q) cosec q
sinq (1 - cos 2 q) tan q (sec 2 q - 1)
Note Above table is applicable only when q Î(0° , 90° ).
Example 2. If sec x + sec2 x = 1, then the value of sin 4 x cos4 x 1
Sol. (b) We have, + =
tan8 x - tan4 x - 2 tan2 x + 1 equal to 2 3 5
sin 4 x (1 - sin 2 x) 2 1
(a) 0 (b) 1 (c) 2 (d) 3 + =
2 3 5
Sol. (c) We have, sec x + sec2 x = 1 6
Þ 3 sin 4 x + 2(1 - sin 2 x) 2 =
sec x = - (sec2 x - 1) = - tan 2 x 5
4 2
Þ sec2 x = tan 4 x Þ1 + tan 2 x = tan 4 x Þ 25 sin x - 20 sin x + 4 = 0
Þ 2
(1 + tan x) = tan x2 8 2
Þ sin 2 x =
2 4 8 5
Þ 1 + 2 tan x + tan x = tan x
Þ cos2 x =
Þ tan 8 x - tan 4 x - 2 tan 2 x = 1
Þ tan 2 x =
Þ tan 8 x - tan 4 x - 2 tan 2 x + 1 = 2
sin 8 x cos8 x () 4 () 4
\ + = +
2 2 8 27 8 27
x + y +1
Example 3. If sin2 q = , then x must be 2 3
2x = +
625 625
(a) - 3 (b) 2 (c) 1 (d) 0
5 1
2 2 = =
x + y +1 625 125
Sol. (c) We have, sin 2 q =
2x
0 £ sin 2 q £ 1 Complementary and
x2 + y 2 + 1
\ 0£
2x
£1 Supplementary Angles
Þ 0 £ x2 + y 2 + 1 £ 2x If the sum of two angles is equal to a right angle, then
Þ 2 2
0 £ x + y - 2x + 1 £ 0 these angles are known as complementary angles of each
Þ x2 - 2x + 1 + y 2 = 0 other. Thus, q and 90° - q are complementary angles of
Þ ( x - 1) 2 + y 2 = 0 each other.
\ x - 1 = 0 and y = 0 Now, if the sum of two angles is equal to two right
x = 1and y = 0 angles, then these angles are known as supplementary
angles of each other.
sin4 x cos4 x 1
Example 4. If + = , then Thus, q and 180° - q are supplementary angles of each
2 3 5
other.
4 sin 8 x cos8 x 1
(a) tan 2 x + (b) + = e. g. , 23° and 67° are complementary angles of each other
3 8 27 125
1 sin 8 x cos8 x 2 while 167° and 13° are supplementary angles of each
(c) tan 2 x = (d) + =
3 8 27 125 other.
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