I have to solve the equation $$y = 12 + 2.6 \cos\left(\frac{x\pi}{4}\right) + 1.2 \sin\left(\frac{x\pi}{4}\right), \space\space\space\space 0\leq x \leq 10$$
when $y=11$. This is so that I can find the range of values of $x$ for which $y>11$.
Now, if I plot the graph, I can see there are two values of $x$ when $y=11$.
However, I tried to solve this using the $t$-formula: $\sin x = \frac{2t}{1+t^2}$ and $\cos x = \frac{1-t^2}{1+t^2}$, where $t = \tan \frac{x}{2}$.
If I use the substitution, the equation reduces to $$y = \frac{9.4t^2 + 2.4t + 14.6}{1+t^2}$$ Now if I set $y=11$, I obtain $t=2.427...$ and $t=-0.927...$.
However, here is the problem. This gives more than two values for $x$ in the range than the graph shows and it doesn't even give the correct values of $x$ when I inverse tan. So what is wrong here?