- Show that \(\theta=18^\circ\) satisfies the equation \(\cos 3\theta=\sin 2\theta\).
If \(\cos\alpha=\sin\beta\), what can we say about \(\alpha\) and \(\beta\)?
By expressing \(\cos 3\theta\) and \(\sin 2\theta\) in terms of trigonometrical functions of \(\theta\), show that \(\sin 18^\circ\) is a root of the equation \[4x^2+2x-1=0...\]
How could we write \(\cos3\theta\) and \(\sin2\theta\) in terms of \(\cos\theta\) and \(\sin\theta\)? Could we say \(\cos(3\theta)=\cos(2\theta+\theta\))?
If you can write \(\sin2\theta-\cos3\theta=0\) in terms of \(\cos\theta\) and \(\sin\theta\), can you see anything that looks similar to \(4x^2+2x-1\)?
… and hence find its value as a surd.
How could we solve a quadratic equation to give an exact solution?
- Find all solutions between \(0^\circ\) and \(360^\circ\) of the equation \[4\tan 2x=\cot x.\]
Can we write \(\tan\) and \(\cot\) in terms of \(\cos\) and \(\sin\)?
Can we learn from what we did for the first part?