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| Inverse Trigonometric Functions |
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Definition of sin-1x
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| For x Î [-1,
1], if q is an angle whose sine is x, then we say
that sine inverse x is q and write sin-1x = q. |
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| For practical purposes, only principal values of sin-1x are considered. |
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| sin-1x is a function of x with domain [-1,1] and range [-p/2, p/2]. |
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| The graph of sin-1x is as shown in the figure below. |
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| Definition of cos-1 x |
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 cosine inverse x is q and write cos-1x = q. |
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| In particular, |
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| For practical purposes, only principal values of cos-1x are considered. |
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| cos-1x is a function of x with domain [-1,1] and range [0, p]. |
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| The graph of cos-1x is as shown in the figure below. |
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Definition of tan-1x
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| tangent inverse of x is q and write tan-1x = q. |
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| In particular, |
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| The graph of tan-1x is as shown in the figure below. |
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Definition of cot-1x
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| For x Î R, if
q is an angle whose cotangent is x, then we say
that cotangent inverse of x is q and write cot-1x= q. |
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| In particular, |
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| For practical purposes, only principal values of cot-1x are considered. |
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Definition of sec-1x
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| For x Î R -
(-1, 1), if q is an angle whose secant is x, then
we say that secant inverse of x is q and write sec-1x = q. |
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| In particular, |
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| For practical purposes, only principal values of sec-1x are considered. |
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| sec-1x is a function of x with domain R-(-1,1) and range [0,p]-{p/2}. |
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Definition of cosec-1x
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| For x Î R -
(-1,1), if q is an angle whose cosecant is x,
then we say that cosecant inverse of x is q and
write cosec-1 x = q. |
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| In particular, if q Î
[-p/2, p/2] -
{0}, then q is called the principal value of
cosec-1 x. |
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| For practical purposes, only principal values of cosec-1x are considered. |
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| cosec-1x is a function of x with domain R-(-1,1) and range [-p/2, p/2]-{0}. |
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| Note: |
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| sin-1 x is only a symbol to denote sine inverse of x. |
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| Table of domain and range of inverse trigonometric function |
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| Relation between inverse functions: |
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| Prove the following identities: |
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| Proof: |
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| Let, |
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| Let, |
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| Let, |
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| Let, |
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| Example 1: |
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| Solve the following equation: |
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| Suggested answer: |
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| Example 2: |
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| Prove the following equation: |
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| Suggested answer: |
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