Trigonometric ratios and Identities

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Session 1
Trigonometric ratios and Identities
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Topics
Measurement of Angles
Definition and Domain and Range of Trigonometric
Function
Compound Angles
Transformation of Angles
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Measurement of Angles
J001
Angle is considered as the figure obtained by
rotating initial ray about its end point.
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Measure and Sign of an Angle
J001
Measure of an Angle -
Amount of rotation from initial side to terminal
side.
Sign of an Angle -
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Right Angle
J001
Revolving ray describes one quarter of a circle
then we say that measure of angle is right angle
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Quadrants
J001
XOX x - axis
YOY y - axis
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System of Measurement of Angle
J001
Measurement of Angle
Circular System or Radian Measure
Sexagesimal System or British
System
Centesimal System or French
System
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System of Measurement of Angles
J001
Sexagesimal System (British System)
Centesimal System (French System)
NO
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System of Measurement of Angle
J001
Circular System
1c
If OA OB arc AB
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System of Measurement of Angle
J001
Circular System
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Relation Between Degree Grade And Radian Measure
of An Angle
J002
OR
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Illustrative Problem
J002
Find the grade and radian measures of the angle
5o3730
Solution
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Illustrative Problem
J002
Find the grade and radian measures of the angle
5o3730
Solution
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Relation Between Angle Subtended by an Arc
At The Center of Circle
J002
Arc AC r and Arc ACB ?
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Illustrative Problem
J002
A horse is tied to a post by a rope. If the horse
moves along a circular path always keeping the
rope tight and describes 88 meters when it has
traced out 72o at the center. Find the length of
rope. Take ? 22/7 approx..
Solution
Arc AB 88 m and AP ?
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Definition of Trigonometric Ratios
J003
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Some Basic Identities
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Illustrative Problem
J003
Solution
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Signs of Trigonometric Function In All Quadrants
J004
In First Quadrant
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Signs of Trigonometric Function In All Quadrants
J004
In Second Quadrant
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Signs of Trigonometric Function In All Quadrants
J004
In Third Quadrant
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Signs of Trigonometric Function In All Quadrants
J004
In Fourth Quadrant
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Signs of Trigonometric Function In All Quadrants
J004
ASTC - All Sin Tan Cos
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Illustrative Problem
J004
Solution
Method 1
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Illustrative Problem
J004
Solution
Method 2
Here x -12, y 5 and r 13
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Domain and Range of Trigonometric Function
J005
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Illustrative problem
J005
Solution
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Trigonometric Function For Allied Angles
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Trigonometric Function For Allied Angles
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Periodicity of Trigonometric Function
J005
If f(xT) f(x) ? x,then T is called period of
f(x) if T is the smallest possible positive
number
Periodicity After certain value of x the
functional values repeats itself
Period of basic trigonometric functions
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Trigonometric Ratio of Compound Angle
J006
Angles of the form of AB, A-B, ABC, A-BC etc.
are called compound angles
(I) The Addition Formula
? sin (AB) sinAcosB cosAsinB
? cos (AB) cosAcosB - sinAsinB
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Trigonometric Ratio of Compound Angle
J006
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Illustrative problem
Solution
(i) Sin 75o sin (45o 30o)
sin 45o cos 30o cos 45o sin
30o
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Trigonometric Ratio of Compound Angle
(I) The Difference Formula
? sin (A - B) sinAcosB - cosAsinB
? cos (A - B) cosAcosB sinAsinB
Note - by replacing B to -B in addition formula
we get difference formula
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Illustrative problem
Solution
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Some Important Deductions
? sin (AB) sin (A-B) sin2A - sin2B cos2B -
cos2A
? cos (AB) cos (A-B) cos2A - sin2B cos2B -
sin2A
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To Express acos? bsin? in the form kcos? or
?sin?
acos? bsin?
Similarly we get acos? bsin? ?sin?
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Illustrative problem
Find the maximum and minimum values of 7cos?
24sin?
7cos? 24sin?
Solution
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Illustrative problem
Find the maximum and minimum value of 7cos?
24sin?
Solution
? Max. value 25, Min. value -25 Ans.
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Transformation Formulae
? Transformation of product into sum and
difference
? 2 sinAcosB sin(AB) sin(A - B)
? 2 cosAsinB sin(AB) - sin(A - B)
? 2 cosAcosB cos(AB) cos(A - B)
Proof - R.H.S cos(AB) cos(A - B)
cosAcosB - sinAsinBcosAcosBsinAsinB
2cosAcosB L.H.S
? 2 sinAsinB cos(A - B) - cos(AB) Note
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Transformation Formulae
? Transformation of sums or difference into
products
By putting AB C and A-B D in the previous
formula we get this result
or
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Illustrative problem
Solution
Proved
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Class Exercise - 1
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Class Exercise - 1
Solution -
Let the coin is kept at a distance r from the eye
to hide the moon completely. Let AB Diameter of
the coin. Then arc AB Diameter AB 2.2 cm
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Class Exercise - 2
Solution -
We have 3A 2A A
Þ tan3A tan(2A A)
Þ tan3A tan3A tan2A tanA tan2A tanA
Þ tan3A tan2A tanA tan3A tan2A tanA
(Proved)
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Class Exercise - 3
If sina sinb and cosa cosb, then
Solution -
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Class Exercise - 4
Solution-
LHS sin20 sin40 sin60 sin80
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Class Exercise - 4
Solution-
Proved.
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Class Exercise - 5
Solution -
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Class Exercise - 5
Solution -
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Class Exercise - 6
The maximum value of 3 cosx 4 sinx 5 is
Solution -
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Class Exercise - 6
The maximum value of 3 cosx 4 sinx 5 is
Solution -
\ Maximum value of the given expression 10.
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Class Exercise - 7
Solution -
???are roots of equatoin (i),
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Class Exercise - 7
Solution -
?sin? and sin? are roots of equ. (ii).
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Class Exercise - 7
(iv)
Solution -
?? and ? be the roots of equation (i),
?cos? and cos? are the roots of equation (iv).
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Class Exercise - 8
If a seca c tana d and b seca d tana c,
then
(a) a2 b2 c2 d2 cd
(c) a2 b2 c2 d2
(d) ab cd
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Class Exercise - 8
If a seca c tana d and b seca d tana c,
then
Solution -
Squaring and adding (i) and (ii), we get
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Class Exercise -9
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Class Exercise -9
Solution -
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Class Exercise -10
Solution -
Consequently, cos(a - b) and sin(a b) are
positive.
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Class Exercise -10
Solution -