Trig Identities and Equations

[a] \( \cos x=\frac{-\sqrt{5}}{3}\)

[b] \( \tan x=-\frac{2}{\sqrt{5}}\)

[c] Finding \(\sin 2x \). 

\( \begin{aligned}
\sin 2 x & =2 \sin x \cdot \cos x \\
& =2\left(\frac{2}{3}\right)\left(\frac{-\sqrt{5}}{3}\right) \\
\sin 2 x & =\frac{-4 \sqrt{5}}{9}
\end{aligned}\)

[d] Finding \(\cos 2x \). 

\(\begin{aligned}
\cos 2 x & =1-2 \sin ^2 x \\
& =1-2\left(\frac{2}{3}\right)^2 \\
& =1-2 \times \frac{4}{9} \\
& =\frac{9-8}{9} \\
\cos 2 x & =\frac{1}{9}
\end{aligned} \)

[e] Finding \(\tan 2x \). 

\( \begin{aligned}
\tan 2 x & =\frac{\sin 2 x}{\cos 2 x} \\
& =\frac{\frac{-4 \sqrt{5}}{9}}{\frac{1}{9}} \\
\tan 2 x & =-4 \sqrt{5}
\end{aligned}\)

[f] Finding \(\sin 4x \). 

\(\begin{aligned}
\sin 4 x & =\sin 2(2 x) \\
& =2 \sin (2 x) \cos (2 x) \\
& =2 \times \frac{-4 \sqrt{5}}{9} \times \frac{1}{9} \\
\sin 4 x & =\frac{-8 \sqrt{5}}{81}
\end{aligned} \)

[h] Finding \(\cos 4x \). 

\( \begin{aligned}
\cos 4 x & =\cos 2(2 x) \\
& =2 \cos ^2 2 x-1 \\
& =2\left(\frac{1}{9}\right)^2-1 \\
& =2 \times \frac{1}{81}-1 \\
\cos 4 x & =-\frac{79}{81}
\end{aligned}\)

                  [i] Finding \(\tan 4x \). 

\( \begin{aligned}
\tan 4 x & =\frac{\sin 4 x}{\cos 4 x} \\
& =\frac{-\frac{8 \sqrt{5}}{81}}{\frac{-29}{81}} \\
\tan 4 x & =\frac{8 \sqrt{5}}{79}
\end{aligned}\)