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NSW Y12 Maths - Extension 1 Trig Functions t Formulae Method

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t Formulae Method Theory

$\begin{aligned} t=\tan \frac{\theta}{2} \qquad \sin \theta & =\frac{2 t}{1+t^2} \\ \cos \theta & =\frac{1-t^2}{1+t^2} \\ \tan \theta & =\frac{2 t}{1-t^2} \end{aligned}$\\ \textbf{Example}\\ Solve \(5 \cos \theta+3 \sin \theta=4\) using the 't' formulae for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\)\\ \textbf{Solution}\\ $\begin{aligned} 5 \times \frac{\left(1-t^2\right)}{1+t^2}+\frac{3 \times 2 t}{1+t^2}&=4 \\ 5-5 t^2+6 t&=4\left(1+t^2\right) \\ 0&=9 t^2-6 t-1 \\ t&=\frac{6 \pm \sqrt{36-4 \times 9 \times-1}}{18} \\ &=\frac{6 \pm 6 \sqrt{2}}{18} \\ t&=\frac{1 \pm \sqrt{2}}{3} \end{aligned}$\\ $\begin{aligned} \tan \frac{\theta}{2}&=\frac{1+\sqrt{2}}{3} \text { or } \tan \frac{\theta}{2}=\frac{1-\sqrt{2}}{3} \\ \theta&=72^{\circ} 39 \text { or } 344^{\circ} 17^{\prime} \end{aligned}$\\

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Videos relating to t Formulae Method.

  • t Formulae Method - Video - Using the t-formulae to solve trig 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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