How do you find tan and cot if cot = 12/5 and sin <0?

May 2, 2015

Easy steps I assume you have to calculate tan and cos

$\tan \theta = \frac{1}{\cot} \theta$
$\tan \theta = \frac{5}{12}$
$\sin \frac{\theta}{\cos} \theta = \frac{5}{12}$
$12 \sin \theta - 5 \cos \theta = 0$
${\sin}^{2} \theta + {\cos}^{2} \theta = 1$
Assume sin theta =x ; cos theta = y
${x}^{2} + {y}^{2} = 1$
$12 x - 5 y = 0$
$y = \frac{12 x}{5}$
${x}^{2} + \frac{144 {x}^{2}}{25} = 1$
${x}^{2} \left(1 + \frac{144}{25}\right) = 1$
${x}^{2} = \frac{1}{\frac{169}{25}}$
${x}^{2} = {\left(\frac{5}{13}\right)}^{2}$
$x = \pm \frac{5}{13}$
$\sin \theta = \pm \frac{5}{13}$
As $\sin \theta < 0 ,$ $\sin \theta = - \frac{5}{13}$
$\cos \theta = - \frac{12}{13}$