NSW Y11 Maths - Extension 1 Inverse Trig Functions and Identities Double Angle Formulae

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NSW Syllabus Reference: ME-T2 Further Trigonometric Identities. This will require student to

derive and use the sum and difference expansions for the trigonometric functions \(\sin (𝐴 \pm 𝐵),\, \cos (𝐴 \pm 𝐵)\) and \(\tan (𝐴 \pm 𝐵)\) (ACMSM044)
derive and use the double angle formulae for \(\sin 2𝐴,\, \cos 2𝐴\) and \(\tan 2𝐴\) (ACMSM044)
derive and use expressions for \(\sin 𝐴,\, \cos 𝐴\) and \(\tan 𝐴\) in terms of \(t\) where \(t=\tan \dfrac{A}{2}\) (the \(t\)-formulae)
derive and use the formulae for trigonometric products as sums and differences for \(\cos 𝐴 cos 𝐵,\, sin 𝐴 sin 𝐵,\, sin 𝐴 cos 𝐵\) and \(cos 𝐴 sin 𝐵\) (ACMSM047)

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Videos relating to Double Angle Formulae.

Double Angle Formulae - Video - Trigonometric Double Angle Identities

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Double Angle Formulae - Video - Identities: double angles

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For all angles A

\(\sin 2A = 2\sin A\cos A\)
\(\cos 2A = {\cos ^2}A - {\sin ^2}A,\,\,{\rm{or}}\,\cos 2A = 2{\cos ^2}A - 1,\,\,{\rm{or }}\cos 2A - 1 - 2{\sin ^2}A\)
\(\tan 2A = \dfrac{{2\tan A}}{{1 - {{\tan }^2}A}}\)

Problems involving the above formulae will be presented in the subtopic