# If tan x = -7/24; and cos x > 0; find all possible trigonometric ratios?

May 13, 2018

#### Explanation:

Here,
 (i)tanx=-7/24 < 0 =>II^(nd)Quadrant orcolor(blue)( IV^(th)Quadrant

$\left(i i\right)$ given that,cosx > 0 =>I^(st) Quadrant or color(blue)( IV^(th) Quadrant

From $\left(i\right) \mathmr{and} \left(i i\right)$ we can say that,

$\frac{3 \pi}{2} < x < 2 \pi \implies \textcolor{b l u e}{I {V}^{t h} Q u a \mathrm{dr} a n t} \implies \cos x > 0 , \sec x > 0$

$\mathmr{and} \sin x < 0 , \csc x < 0 , \tan x < 0 , \cot x < 0$

$\left(a\right) \sec x = \sqrt{\left(1 + {\tan}^{2} x\right)} = \sqrt{1 + \frac{49}{576}} = \sqrt{\frac{625}{576}} = \frac{25}{24}$

$\left(b\right) \cos x = \frac{1}{\sec} x = \frac{1}{\frac{25}{24}} = \frac{24}{25}$

$\left(c\right) \sin x = - \sqrt{1 - {\cos}^{2} x} = - \sqrt{1 - \frac{576}{625}} = - \frac{7}{25}$

$\left(d\right) \csc x = \frac{1}{\sin} x = \frac{1}{- \frac{7}{25}} = - \frac{25}{7}$

$\left(e\right) \cot x = \frac{1}{\tan} x = \frac{1}{- \frac{7}{24}} = - \frac{24}{7}$