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NSW Y12 Maths - Extension 1 Trig Functions Compound Angle Equations

Resources for Compound Angle Equations

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Compound Angle Equations Theory

\textbf{Compound-angle formulae:}\\ $\begin{aligned} & \sin (A+B)=\sin A \cos B+\cos A \sin B \\ & \sin (A-B)=\sin A \cos B-\cos A \sin B \\ & \cos (A+B)=\cos A \cos B-\sin A \sin B \\ & \cos (A-B)=\cos A \cos B+\sin A \sin B \\ & \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} \\ & \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} \end{aligned}$\\ \textbf{Double-angle formulae:} \\ $\begin{aligned} & \sin 2 A=2 \sin A \cos A \\ & \cos 2 A=\cos ^2 A-\sin ^2 A \\ & \tan 2 A=\frac{2 \tan A}{1-\tan ^2 A} \end{aligned}$\\ \begin{multicols}{2} \textbf{Example 1}\\ In the domain \(-\pi \le \theta \le \pi\), find the solutions to \(2 \cos 2 \theta = 1-4 \cos \theta\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} &2\left(2 \cos ^2 \theta-1\right)=1-4 \cos \theta \\ \end{aligned}$\\ $\begin{aligned} &\text { Let } x=\cos \theta \\ \end{aligned}$\\ $\begin{aligned} 2\left(2 x^2-1\right)&=1-4 x \\ 4 x^2-2&=1-4 x \\ \end{aligned}$\\ $\begin{aligned} 4 x^2+4 x-3&=0\\ (2 x+3)(2 x-1)&=0 \\ \end{aligned}$\\ \\ $\begin{aligned} x=-\frac{3}{2}, x=\frac{1}{2} \quad\left(x \neq-\frac{3}{2}\right) \\ \end{aligned}$\\ \\ $\begin{aligned} \cos \theta&=\frac{1}{2} \\ \therefore \theta&=\frac{\pi}{3} \text { or }-\frac{\pi}{3} \end{aligned}$\\

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Videos relating to Compound Angle Equations.

  • Compound Angle Equations - Video - Using Double Angle Identities to Solve Equations

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NSW Year 12 Maths Extension 1 - Trig Equations

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NSW Year 12 Maths Extension 1 - Proof

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NSW Year 12 Maths Extension 1 -Vectors

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NSW Year 12 Maths Extension 1 - Calculus

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NSW Year 12 Maths Extension 1 -Differential Equations

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NSW Year 12 Maths Extension 1 -Projectile Motion

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NSW Year 12 Maths Extension 1 -Binomial Distribution

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