Given:
[tex]5tan^2\theta+5tan\theta=0[/tex][tex]-\pi\leq\theta<\pi[/tex]
To determine the value(s) for θ based on the given equation, we first let tan θ equal to u. So,
[tex]\begin{gathered} 5tan^{2}\theta+5tan\theta=0 \\ 5u^2+5u=0 \\ Simplify\text{ and rearrange} \\ 5u(u+1)=0 \end{gathered}[/tex]
Next, we get the value of u when 5u=0 or u+1=0:
[tex]\begin{gathered} 5u=0 \\ u=0 \end{gathered}[/tex][tex]\begin{gathered} u+1=0 \\ u=-1 \end{gathered}[/tex]
This means that:
[tex]tan\theta=0,\text{ }tan\theta=-1[/tex]
Now we consider the given range above. Hence,
For tan θ=0:
[tex]\theta=0,\text{ }\theta=-\pi[/tex]
For tan θ=-1:
[tex]\theta=\frac{3\pi}{4},\theta=-\frac{\pi}{4}[/tex]
Therefore, the answers are:
[tex]\begin{gathered} 0 \\ -\pi \\ -\frac{\pi}{4} \\ \frac{3\pi}{4} \end{gathered}[/tex]
