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I have been revising trigonometry and I couldn't solve this particular one from my book. One has to find the value of given expression $\frac{tan\alpha}{1-cos\alpha}$, if $sin\alpha=-\frac{2}{3}$ and $ctn\alpha>0$.

I tried expressing every trigonometric function through $sin\alpha$ by multiplying numerator and denominator by $cos\alpha$. That would yield $sin\alpha$ in numerator and $cos\alpha$ functions in denominator.

Then I tried expanding further:

Since it was given that $ctan\alpha$>0, then $ctn\alpha=\frac{cos\alpha}{sin\alpha} \Rightarrow cos\alpha=ctn\alpha \times sin\alpha \Rightarrow \pm \sqrt{\frac{1}{sin^2\alpha}-1} \times sin\alpha= \sqrt{\frac{1}{sin^2\alpha}-1} \times sin\alpha$

I suppose that the value of $ctn\alpha$ will have to be positive.

$\frac{sin\alpha}{cos\alpha+cos^2\alpha}=\frac{sin\alpha}{\sqrt{\frac{1}{sin^2\alpha}-1} \times sin\alpha-1+sin^2\alpha}$

When I input initial condition for sine value, I still somehow can't get the result stated in my book (answers are given in random order; my answer doesn't quite match any of them)

I would appreciate any help with this!

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  • $\begingroup$ Something that looks suspect in your work. $\cos^2 = 1-\sin^2 a$ regardless of the quadrant of $a.$ You have a sign flipped in your last expression. $\endgroup$ – Doug M Mar 30 '17 at 17:16
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HINT:

As $\dfrac{\cos\alpha}{\sin\alpha}=\cot\alpha>0,\cos\alpha<0\implies\cos\alpha=-\sqrt{1-\sin^2\alpha}$

$\tan\alpha=\dfrac{\sin\alpha}{\cos\alpha}=?$

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The given question is $sin\alpha = - \frac{2}{3}$ and ctn\alpha >o$ This implies that $sin\alpha = -\frac{y}{r} = -frac{2}{3} hence using the Phythagoras theorm one can find that $y=-2,x=-\surd5$ Hence $$tan\alpha = \frac{2}{\surd5},cos\alpha=-\frac{\surd5}{3}$$ putting these values in given expression you would get $$\frac{tan\alpha}{1-cos\alpha}=\frac{6}{5+3\surd5}$$

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Thanks, both respondents! I don't know why I've overcomplicated this so much for myself. With both methods, I finally managed to get the result of $\frac {3(3 \sqrt {5}+5)}{10}$, which matches the given set of answers.

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