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

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2.5 Inverse Trigonometric Functions
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• LEARNING OUTCOMES
• By the end of the lesson, students should be able
  to
• define and understand the inverses of
  trigonometric functions.
• sketch the graphs of trigonometric functions and
  their inverses

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2.5 (a) and (b) The Inverse of Trigonometric
Functions
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• Note
• Inverse function is valid if the function is
  one-to-one mapping
• sin-1x, cos-1x and tan-1x can be defined for a
  restricted domain. These domains are the values
  of x for which the sine, cosine and tangent
  mappings are one-to-one.
• For f(x) sin x and to be one-to-one mapping,
  
• and Rf -1,1.
• Hence, the inverse mapping, f -1(x) sin-1 x
  maps -1,1 onto .

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• d) For f(x) cos x to be one-to-one mapping,
  
• and Rf -1,1.
• Hence the inverse mapping, f -1 (x) cos-1
  x maps
• - 1,1 onto 0, .
• For f(x) tan x to be one-to-one mapping,
• and Rf -?,?.
• Hence, the inverse mapping f -1 (x) tan-1 x
  maps(-? ,?) onto .

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• Graph of y sin-1x
• domain -1 , 1
• range

y sin -1 x
y sin x
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2) Graph of y cos-1 x domain -1 ,
1?? range 0 , ???
y cos-1 x
y cos x
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3) Graph of y tan-1x domain -? ,?
range asymptote y
y tan x
y tan-1 x
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• Example 1
• Find the value of
• (a) sin ( sin-1 0.7) (b) cos-1
• (c) tan-1 ( -1 )

Solution a) sin ( sin-1 0.7 ) 0.7 b)
cos-1

c) Let y tan-1 ( -1 ) , then tan y -1

- tan

y
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• Example 2
• Find the value without using the calculator
• cos b) sin

Solution a) Let y sin-1
b) Let y cos-1
cos y
sin y
- sin
cos


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b) Let




tan x 1



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• Example 3
• Show that
• cos ( 2 tan -1x ) b) tan -1
• c) cos

cos ( 2 tan-1x ) cos 2y 2 cos2y 1
2 2
Solution a) Let y tan-1 x ,
y
tan y x
y
x
1
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c) Let y sin-1x ,


(apply cos2y sin2y 1 )