Question

Use the double angle formulae of cosine to expand the following:

(1) \(\cos 6 \mathrm{~A}\)

(2) \(\cos 80^{\circ}\)

(3) \(\cos \mathrm{A}\)

Answer 1

(1)
\[
\begin{aligned}
& \cos 6 A=\cos [2(3 A)] \\
& \therefore \cos 6 A=\cos ^2 3 A-\sin ^2 3 A \\
& \therefore \cos 6 A=2 \cos ^2 3 A-1 \\
& \therefore \cos 6 A=1-2 \sin ^2 3 A
\end{aligned}
\]

 

(2)
\[
\begin{aligned}
& \cos 80^{\circ}=\cos \left[2\left(40^{\circ}\right)\right] \\
& \therefore \cos 80^{\circ}=\cos ^2 40^{\circ}-\sin ^2 40^{\circ} \\
& \therefore \cos 80^{\circ}=2 \cos ^2 40^{\circ}-1 \\
& \therefore \cos 80^{\circ}=1-2 \sin ^2 40^{\circ}
\end{aligned}
\]

 

(3)
\[
\begin{aligned}
& \cos A=\cos \left[2\left(\frac{A}{2}\right)\right] \\
& \therefore \cos A=\cos ^2 \frac{A}{2}-\sin ^2 \frac{A}{2} \\
& \therefore \cos A=2 \cos ^2 \frac{A}{2}-1 \\
& \therefore \cos A=1-2 \sin ^2 \frac{A}{2}
\end{aligned}
\]