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NSW Y11 Maths Standard NSW Y11 Maths Advanced NSW Y11 Maths Extension 1

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Functions Polynomials Inverse Trig Functions and Identities Rates of Change Permutations and Combinations

NSW Y11 Maths - Extension 1 Inverse Trig Functions and Identities Inverse Trig Functions

Resources for Inverse Trig Functions

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Inverse Trig Functions Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-T1 Inverse Trigonometric Functions. This will require student to 

  • define and use the inverse trigonometric functions (ACMSM119)
  • sketch graphs of the inverse trigonometric functions
  • use the relationships \(\sin(\sin ^{-1}x)=x\) and \(\sin^{-1}(\sin x) =x\), \(\cos(\cos ^{-1}x)=x\) and \(\cos^{-1}(\cos x) =x\), \(\tan(\tan ^{-1}x)=x\) and \(\tan^{-1}(\tan x) =x\) where appropriate, and state the values of \(x\) for which these relationships are valid
  • prove and use the properties: \(\sin^{-1}(-x)=-\sin^{-1}x,\, \cos^{-1}(-x)=\pi -\cos^{-1}x,\, \tan^{-1}(-x)=- \tan ^{-1}x\) and \(\cos^{-1}x+ \sin^{-1}x=\dfrac{\pi}{2}\)
  • solve problems involving inverse trigonometric functions in a variety of abstract and practical situations

Ref: https://educationstandards.nsw.edu.au/

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Videos relating to Inverse Trig Functions.

  • Inverse Trig Functions - Video - Inverse Trig Functions With Double Angle Formulas and Half Angle Identities

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Theory

Corresponding to each of the six trigonometric functions \(y = \sin x\), \(y = \cos x\), \(y = \tan x\), \(y = cosec x\), \(y = \sec x\) and \(y = \cot x\) there are six associated inverse trigonometric functions.

In general, if \(y = f(x)\) is a function, then for each \(x\) in the domain, there is one and only one value of \(y\). The relation obtained by interchanging \(x\) and \(y\) is \(x = f(y)\). If for each \(x\) in the domain, \(y\) is uniquely determined, then this represents a new function, called the inverse function to \(y = f(x)\) and is denoted by \(y = {f^{ - 1}}(x)\).

To obtain the inverse function \(y = {f^{ - 1}}(x)\) from \(y = f(x)\), \(x\) and \(y\) are interchanged and consequently, so are the domains and ranges.

The domain of \(y = {f^{ - 1}}(x)\) is the range of \(y = f(x)\) and the range of \(y = {f^{ - 1}}(x)\) is the domain of \(y = f(x)\).

The notation used for the inverse trigonometric functions are

  • If \(y = \sin x\) then the inverse function is \(y = {\sin ^{ - 1}}x\)

  • If \(y = \cos x\) then the inverse function is \(y = {\cos ^{ - 1}}x\)

  • If \(y = \tan x\) then the inverse function is \(y = {\tan^{ - 1}}x\)

  • If \(y = cosec x\) then the inverse function is \(y = {cosec ^{ - 1}}x\)

  • If \(y = \sec x\) then the inverse function is \(y = {\sec ^{ - 1}}x\)

  • If \(y = \cot x\) then the inverse function is \(y = {\cot ^{ - 1}}x\)