# Given theta is an acute angle such that sin theta = 5/13 find the value of tan (theta - pi4)

Given theta is an acute angle such that sin theta = $\frac{5}{13}$ find the value of $\mathrm{tan}\left(\theta -\pi /4\right)$

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Fatema Sutton

$\mathrm{tan}\left(\theta -\frac{\pi }{4}\right)=\left(\mathrm{tan}\theta -\frac{\mathrm{tan}\left(\frac{\pi }{4}\right)}{1+\mathrm{tan}\theta \frac{\mathrm{tan}\pi }{4}}\right)=$
$\frac{\mathrm{tan}\theta -1}{1+\mathrm{tan}\theta }.$
In a right triangle with hypotenuse length 13 and a leg length of 5, the other leg has a length $\sqrt{{13}^{2}-{5}^{2}}=\sqrt{144}=12$. In this triangle, when $\mathrm{sin}\theta =\frac{5}{13}$ (opp/hyp) then $\mathrm{tan}\theta =op\frac{p}{a}dj=\frac{5}{12}$.
Therefore $\mathrm{tan}\left(\theta -\frac{\pi }{4}\right)=\frac{\frac{5}{12}-1}{1+\frac{5}{12}}=\frac{-\frac{7}{12}}{\frac{17}{12}}=-\frac{7}{17}.$