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In class we've learned 2 methods for solving trigonometric equations.

  1. T-Formula: If you're not familiar with the T-Formula look here; T-Formula
  2. Auxiliary Method: If you're not familiar look here

When I was trying to solve the equation $\sin(\theta) + \cos(\theta) = -1$, where $0 \leq \theta \leq 360$. When I used the auxiliary method I get the 2 answers of $180, 270$. But when I use the T-Formula I only get $270$. In the T-Formula I end up getting to the following equation:

$$\frac {2t + 1 - t^2}{1+t^2} = -1$$ $$2t+1-t^2 = -1-t^2$$ $$2t=-2$$ $$\therefore t=-1$$ But $t=\tan(\frac{\theta}{2})$, so... $$\tan(\frac {\theta}{2}) = -1$$

But there $\theta \neq 180$ as $\tan(90)$doesn't exist. But that is one of the answers.

What am I doing wrong? Why doesn't the T-Formula get all the answers (it should)?

frog1944
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  • I think you're not taking both the roots when solving for $\tan \frac{\theta}{2}$. there's a $\pm$ involved that you're missing I believe. I wish you'd show your work so I can comment more – Siddharth Bhat May 06 '16 at 10:55
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    Read the last paragraph of the link you posted: "This implies that some solutions are missing. Hence when t-formula is used, we have to check whether are solutions.We can do this by substituting into the equation." – N74 May 06 '16 at 10:57
  • Ok, thank you very much. My teacher didn't tell us this – frog1944 May 06 '16 at 11:00
  • Why did this get marked as an exact duplicate? The "duplicate" asks about a different question and solves it using different methods (which is the problem of my question, the method isn't working) – frog1944 May 06 '16 at 11:05

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