Simplify \({9}^{2}\) to \(81\).
\[81={12}^{2}+{6}^{2}-2\times 12\times 6\cos{y}\]
Simplify \({12}^{2}\) to \(144\).
\[81=144+{6}^{2}-2\times 12\times 6\cos{y}\]
Simplify \({6}^{2}\) to \(36\).
\[81=144+36-2\times 12\times 6\cos{y}\]
Simplify \(2\times 12\times 6\cos{y}\) to \(144\cos{y}\).
\[81=144+36-144\cos{y}\]
Simplify \(144+36-144\cos{y}\) to \(-144\cos{y}+180\).
\[81=-144\cos{y}+180\]
Subtract \(180\) from both sides.
\[81-180=-144\cos{y}\]
Simplify \(81-180\) to \(-99\).
\[-99=-144\cos{y}\]
Divide both sides by \(-144\).
\[\frac{-99}{-144}=\cos{y}\]
Two negatives make a positive.
\[\frac{99}{144}=\cos{y}\]
Simplify \(\frac{99}{144}\) to \(\frac{11}{16}\).
\[\frac{11}{16}=\cos{y}\]
Switch sides.
\[\cos{y}=\frac{11}{16}\]
Ask: What values of \(y\) will make \(\cos{y}\) equal \(\frac{11}{16}\)?
\[y=\cos^{-1}{(\frac{11}{16})},2\pi -\cos^{-1}{(\frac{11}{16})}\]
Since cos is a periodic function, add the periodicity.
\[\begin{aligned}&y=2\pi n+\cos^{-1}{(\frac{11}{16})},n \in Z\\&y=2\pi n+2\pi -\cos^{-1}{(\frac{11}{16})},n \in Z\end{aligned}\]
y=sequence(2*PI*n+arccos(11/16),in(n,Z)),sequence(2*PI*n+2*PI-arccos(11/16),in(n,Z))
