Review question

Can we find the value of $\sin 18^\circ$ as a surd? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6018

Suggestion

1. 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?

1. 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?