Trig Equations using t-formulae (Edexcel A Level Further Maths: Further Pure 1): Revision Note

You must learn the t-formulae for $\sin \theta$, $\cos \theta$ and $\tan \theta$ as they are not given in the formulae booklet!

The t-formulae can't find solutions that make the t-substitution undefined, e.g. $t=\tan \frac{\theta }{2}$ would not pick up a solution at $\theta =\pi$.

Use the substitution $t=\tan \frac{x}{2}$ to solve for $0<x<2\pi$

$2\cos x+\sin x=1$

giving answers to 3 significant figures where necessary.

Answer:

Write $\cos x$ in terms of its t-formula

$\cos x=\frac{1-t^2}{1+t^2}$

Write $\sin x$ in terms of its t-formula

$\sin x=\frac{2t}{1+t^2}$

Substitute these into the equation

$2\left(\frac{1-t^2}{1+t^2}\right)+\frac{2t}{1+t^2}=1$

Multiply both sides by $1+t^2$

$2\left(1-t^2\right)+2t=1+t^2$

Make one side zero

$\begin{array}{rcl}2-2t^2+2t& =& 1+t^2\\ 0& =& 3t^2-2t-1\end{array}$

Solve the quadratic equation, e.g. by factorisation

$\begin{array}{rcl}0& =& \left(3t+1\right)\left(t-1\right)\\ t& =& -\frac{1}{3}\mathrm{or}t=1\end{array}$

Convert each $t$ solution back into $x$ solutions using $t=\tan \frac{x}{2}$

For $\begin{array}{rcl}t& =& -\frac{1}{3}\end{array}$

$\begin{array}{rcl}\tan \frac{x}{2}& =& -\frac{1}{3}\\ \frac{x}{2}& =& \arctan \left(-\frac{1}{3}\right)+n\pi \\ \frac{x}{2}& =& -0.32175...,2.81984...,....\\ x& =& -0.64350...,5.63968...,....\end{array}$

• Find $x$ solutions in the range $0<x<2\pi$

$\begin{array}{rcl}x& =& 5.63968...\end{array}$

For $\begin{array}{rcl}t& =& 1\end{array}$

$\begin{array}{rcl}\tan \frac{x}{2}& =& 1\\ \frac{x}{2}& =& \arctan \left(1\right)+n\pi \\ \frac{x}{2}& =& \frac{\pi }{4},\frac{5\pi }{4},...\\ x& =& \frac{\pi }{2},\frac{5\pi }{2},...\end{array}$

• Find $x$ solutions in the range $0<x<2\pi$

$\begin{array}{rcl}x& =& \frac{\pi }{2}\end{array}$

Present all the solutions (rounding 5.63968... to 3 s.f.)

$\begin{array}{rcl}x& =& 5.64\left(3\mathrm{s}.\mathrm{f}.\right)\mathrm{or}x=\frac{\pi }{2}\end{array}$

Other methods to solve this equation, e.g. writing in the form $R\cos \left(x+\alpha \right)$, score no marks if the question asks you to use t-substitutions.