TOPIC : 2'0 TRIGONOMETRIC FUNCTIONS SUBTOPIC : 2'2 Trigonometric Ratios and Identities

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2.2 Trigonometric Ratios and Identities
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Generally the diagram above can be formulated as
the diagram below
Q
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2.2a Trigonometric Ratios of sin ,cos , tan
, cosec ,sec and cot
?
?
?
?
?
?
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For any acute angle ?, there are six
trigonometric ratios, each of which is defined by
referring to a right angled triangle containing
?.
From the diagram
?
900-
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From the diagram


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Example 1

If , and is an acute angle,
find sec and
?
Solution
5
4
?
3
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Example 2 Given cos x 0.8, evaluate 5 sin x
3 tan x 3 cosec x.
Solution
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Example 3 Given ,
find a)
Solution


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(No Transcript)
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2.2 b Relationship of sin ?, cos ? and tan ?
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900 - ?
z
y
?
Therefore sin (900 -
?) cos ?
x
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• b) cos (900 - ?) sin ?
• Therefore
• cos (900 - ?) sin ?

cot ?
c) tan (900 - ?)

Therefore tan (900 - ?)
cot ?
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2.2 c Trigonometric Ratios of Particular Angles
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Equilateral triangle of sides 2 unit in length


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Isosceles triangle
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The values of trigonometric ratio for some
particular angles are as follows
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• Example 1
• Prove that

Solution
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Example 2Prove thatsec 300 tan 600 sin 450
cosec 450 cos 300 cot 600
Solution
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2.2 d Trigonometric Ratio for Any Angle

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i) Positive Angle
First Quadrant

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Second Quadrant
sin (1800-?) sin ? cos (1800-?) - cos
? tan (1800-?) - tan ?

x
Third Quadrant
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Fourth Quadrant
We can summarize that all the trigonometric
ratios are positive in the first quadrant, sine
is positive in second quadrant, tangent is
positive in the third quadrant and cosine is
positive in the fourth quadrant as shown below.







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ii) Negative Angle

• The rotating arm will describe a negative angle
  if it rotates in a clock wise direction. To
  convert a negative angle to a normal basic angle,
  add 3600 or a multiple of 3600. The value of a
  trigonometrical ratio
• of any negative angle can then be found.

Example An angle of - 400 is equivalent to a
basic angle of - 400 3600 3200. sin
(-400) sin (3200) - sin (3600-3200) - sin 400
cos (-400) cos (3200) cos
(3600-3200) cos 400 tan (-400) tan
(3200) - tan (3600-3200) - tan 400
Then
Hence in general sin (-?) -sin ? cos (-?)
cos ? tan (-?) -tan ?
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• Example 1
• State the trigonometric ratio in acute angle.
• a) sin 1600 b) cos 2200 c)
  tan 3100
• d) cos 1720 e) tan 2460

Solution a) sin 1600
b) cos 2200 sin (1800 200)
cos (1800
400) sin 200
- cos 400 c) tan 3100
d) cos 1720
tan (3600 500)
cos (1800 80) - tan 500
- cos 80
e) tan 2460 tan (1800 660)
tan 660
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• Example 2
• State the trigonometric ratio in acute angle
• a) sin (-250) b) cos (-400)
  c) tan (-480)
• d) sin (-1280) e) cos (-1520)
  f) tan (-1630)

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• d) sin (-1280)
• - sin 520
• e) cos (-1520)
• - cos 280

1280
f) tan (-1630) - tan 170
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• Example 3
• Find the following
• a)sin 1200 b) cos 2100
  c) tan 2250
• d) tan (-1200) e) sin (-1350)
  f) cos (-2700)

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2.2 e Trigonometric Identities
For any position of OP a right angled can be
drawn for which,
x2
y2 OP2 r2
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• Dividing through by r2 becomes
• Since
• cos and
  
• Therefore
• cos2 ? sin2 ? 1

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• Dividing through by x2 becomes
• 1

Since
and
Therefore 1 tan2 ? sec2 ?
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• Dividing through by y2 becomes
• Since and
• Therefore
• cot2 ? 1
  cosec 2 ?

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• Useful formulae and identities are summarized in
  the following diagram.The diagonals show the
  reciprocals of the various trigonometric
  ratios
• The shaded triangles show the three basic
  identities. The sum of the square of the two
  bases is equal to the square of the downward
  vertex.
• i.e sin2? cos2 1 ,1 tan2? sec2?,1 cot2?
  cosec2?

1

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Example 1Prove the identity tan ? cot ? sec
? cosec?

• Solution
• LHS tan ? cot ?

(since sin2 ? cos2 ? 1)
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Example 2Prove the identity(1 sin ? cos ?
)2 2(1 sin ?)(1 cos ? )
Solution

• LHS (1- sin ? cos ?)(1- sin ? cos ?)
• 1-2 sin? 2 cos? sin2? cos2? 2
  sin?cos ?
• 2 2 sin?2cos?2sin?cos? (since
  sin2?cos2?1)
• 2 (1- sin? cos? sin? cos?)
• 2 (1- sin? )(1 cos? ) RHS

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Example 3Prove the identity

• Solution