LESSON NOTE
TOPIC/ LESSON:-TRIGONOMETRIC FUNCTIONS
CLASS:- XI A
GIST OF THE LESSON/ CHAPTER:-
Positive and negative angles.Measuring angles in radians & in degrees and conversion from one
measure to another.Definition of trigonometric functions with the help ofunit circle. Truth of the
identity sin x + cos x=1, for all x. Signs of trigonometricfunctions and sketch of their graphs.
Expressing sin (x+y) and cos (x+y) in terms ofsinx, siny, cosx&cosy. Deducing the identities like
the following:
tan 𝑥±tan 𝑦
tan(𝑥 ± 𝑦) = 1∓𝑡𝑎𝑛𝑥𝑡𝑎𝑛𝑦 , cot(𝑥 ± 𝑦) =
1
(𝑐𝑜𝑡𝑥𝑐𝑜𝑡𝑦∓1 )
𝑐𝑜𝑡𝑦±𝑐𝑜𝑡𝑥
1
cos 𝛼 + cos 𝛽 = 2 cos 2 (𝛼 + 𝛽) cos 2 (𝛼 − 𝛽),
1
1
sin 𝛼 + sin 𝛽 = 2 sin 2 (𝛼 + 𝛽) cos 2 (𝛼 − 𝛽),
1
1
cos 𝛼 − cos 𝛽 = −2 sin 2 (𝛼 + 𝛽) sin 2 (𝛼 − 𝛽),
1
1
sin 𝛼 − sin 𝛽 = 2 cos 2 (𝛼 + 𝛽) sin 2 (𝛼 − 𝛽),
Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of
trigonometric equations of the type sinӨ= sin ∝, cosӨ= cos∝and tanӨ= tan∝
EXPLANATION:
RADIANS AND ANGLES IN STANDARD POSITION
Degree Measure of an Angle:
Angles can be measured in degrees where 360 is one complete rotation.
A rotation angle is formed by rotating an initial arm through an angle about a fixed point (the vertex).
A positive angle results from a counter clockwise rotation.
A negative angle results from a clockwise rotation.
To place an angle in standard position, the initial ray must be the positive x-axis, the vertex must be at the
origin of the cartesian plane, and the other ray forming the angle is called the terminal angle.
Angles with the same terminal arm are called coterminal angles.
The smallest positive angle in a set of coterminal angles is called the principle angle, the principle angle
will always have a measure between 0 and 360
Ex. 20,1100,380, 340, 700 are all coterminal, the principle angle is 20
Reciprocal Trigonometric Ratios:
cosecant ratio  csc 
1
sin 
cotangent ratio  cot  
secant ratio  sec 
1
cos
1
tan 
These are reciprocals of the primary trigonometric ratios.
The primary and reciprocal ratios can be given in terms of x, y and r.
sin  
y
r
csc 
r
y
cos  
x
r
sec  
r
x
tan  
y
x
cot  
x
y
Radians:
The radian measure of an angle is a ratio that compares the length of an arc of a circle to the radius of the
circle.
Conversion Chart
Degrees to Radians multiply by

180
Radians to Degrees multiply by
180

Arc Length:
a  R
CAST Rule:
Sine ratio
positive positive
Tangent
positive
ratio positive
wherea= arc length
R = radius
 = angle in radians
All ratios
The reciprocal trigonometric ratios
follow the same framework as their
corresponding primary ratio.
Cosine ratio
Graphs of Trigonometric Functions
Sine
Period = 2 
Cosine
Period = 2 
y = a sin (bx + c)
amplitude = a
y = a cos (bx + c)
amplitude = a
period =
period =
2
b
phase shift =
c
b
2
b
phase shift =
c
b
one cycle can be found by solving:
one cycle can be found by solving:
Tangent
Period = 
x –intercepts at  n
Cotangent
Period = 
0  bx  c  2
vertical asymptotes at x =

2
 n
y = a tan (bx + c)

period =
b
phase shift =
0  bx  c  2
x –intercepts at
successive vert. asymptotes for one branch:
  2  bx  c   2
2
 n
vertical asymptotes at x =  n
y = a cot (bx + c)
period =
c
b


b
phase shift =
c
b
successive vert. asymptotes for one branch:
0  bx  c  
Cosecant
Period = 2 
Vertical asymptotes at x =  n
Secant
Period = 2 
Vertical asymptotes at x =
y = a csc (bx + c)
2
 n
y = a sec (bx + c)
2
period =
b
phase shift =

period =
c
b
2
b
phase shift =
one cycle can be found by solving:
c
b
one cycle can be found by solving:
0  bx  c  2
To graph y = a csc (bx + c):
First graph y = a sin (bx + c); draw the
vertical asymptotes at the x-intercepts;
vertical asymptotes at the x-ints,
take the reciprocals.

3
 bx  c 
2
2
To graph y = a sec (bx + c):
First graph y = a cos (bx + c); draw the
vertical asymptotes at the x-intercepts;
vertical asymptotes at the x-ints,
take the reciprocals.
Summary:
period
y = sin x
y = cos x
2
2
x-intercepts
y = tan x

 n
2
n
y = cot x


y = sec x
2
none
n

2
 n
none
2
Some Common Basic Identities:
y = csc x
y-intercepts
0
1
Vertical asymptotes
none
none
0

 n
2
x  n
none
1
none
x

 n
2
x  n
x
tan x 
sin x
cos x
sin2x + cos2x = 1 (Corollaries: sin2x = 1 - cos2x,
sin (2x) = 2 sin xcosx
cos (2x) = cos2x - sin2x = 2cos2x - 1 = 1 - 2sin2x.


sin  x    cos x
2



cos x    sin x
2

cos2x = 1 - sin2x)
(shift identities)
sin(-x) = -sin(x), tan(-x) = -tan(x)
(sine and tangent are odd functions)
cos(-x) = cos(x) (cosine is an even function)
Right Triangles:
sin  
opp
adj
opp
; cos  
; tan  
hyp
hyp
adj
"Soh-Cah-Toa". Remember, cscx = 1/sin x, secx = 1/cosx, cot x = 1/tan x.
Use Pythagoras' formula to determine unknown sides in a right triangle.
TRIGONOMETRIC IDENTITES
Identity
SinA=
CosA=
TanA=
CotA=
SecA=
CscA=
Letter from Triangle
a/c
b/c
a/b
b/a
c/b
c/a
Position on Triangle
Opposite/ Hypotenuse
Adjacent/ Hypotenuse
Opposite/ Adjacent
Adjacent/ Opposite
Hypotenuse/ Adjacent
Hypotenuse/ Opposite
The Pythagorean Identity for sines and cosines is:
1. sin2x + cos2x = 1
2. 1 + cot2x = csc2x
3. 1 + tan2x = sec2x
Sum and Difference Identities
sin(u  v)  sin u cos v  cos u sin v sin(u  v)  sin u cos v  cos u sin v
cos(u  v)  cos u cos v  sin u sin v cos(u  v)  cos u cos v  sin u sin v
tan(u  v) 
tan u  tan v
tan u  tan v
tan(u  v) 
1  tan u tan v
1  tan u tan v
Double Angle
and
Power Reducing Identities
sin(2u )  2sin u cos u sin 2 u 
1  cos  2u 
cos(2u )  cos 2 u  sin 2 u
 2cos 2 u  1
cos 2 u 
2
1  cos  2u 
 1  2sin 2 u
tan(2u) 
2tan u
1  tan u
2
tan 2 u 
2
1  cos  2u 
1  cos  2u 
Half Angle Identities
1  cos u
1  cos u
sin u
u
u
 u  1  cos u
sin    
cos    
tan   

2
sin
u
1

cos u
2
2
2
2
 
 
 
Product to Sum
1
1
 cos(u  v)  cos(u  v)  cos u cos v   cos(u  v)  cos(u  v) 
2
2
1
1
sin u cos v   sin(u  v)  sin(u  v)  cos u sin v   sin(u  v)  sin(u  v) 
2
2
sin u sin v 
Sum to Product
uv
u v
uv uv
sin u  sin v  2sin 
 cos 
 sin u  sin v  2cos 
 sin 

2
2




 2   2 
uv
u v
u v u v
cos u  cos v  2cos 
 cos 
 cos u  cos v  2sin 
 sin 

 2 
 2 
 2   2 
Law of Cosines:
Lower case a, b, c are always sides, Capital A, B, C are angles. A is opposite a, etc.
c 2  a 2  b 2  2ab cos C . Analogous formulas for a andb.
Find the side opposite the largest given angle first, if possible. Beware of ambiguous cases.
Law of Sines:
sin A sin B sin C


a
b
c
Make sure you're in the right mode (degree/radian).
ADDITIONAL PROBLEMS:
1. Prove
sin 3x  3 sin x  4 sin 3 x
2. Verify the identity: sec x  cos x  sin x tan x
3. Solve 3 tan 2 x  1  0 .
4. Solve cos 3 x  cos x
5.
6.
5
in the 1st quadrant find the other five trigonometric function values.
13
Solve the following equations.
Given sin  
a)
3 cos   cos tan 
b) sec x csc x  2csc x  0
c) csc2   cot 2   3
HOME ASSIGNMENTS:
Exercises of NCERT Text Book: Ex 3.1, 3.2, 3.3, 3.4,Miscellaneous Exercise Chapter 3