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

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#### Explanation

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#### Explanation:

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Jim G. Share
Apr 19, 2018

$\text{see explanation}$

#### Explanation:

$\text{using the "color(blue)"trigonometric identities}$

•color(white)(x)tan^2x+1=sec^2x

$\Rightarrow \sec x = \pm \sqrt{{\tan}^{2} x + 1}$

•color(white)(x)sin^2x+cos^2x=1

$\Rightarrow \sin x = \pm \sqrt{1 - {\cos}^{2} x}$

$\text{Given "tanx<0" and "cosx>0" then}$

$\text{x is in the fourth quadrant}$

$\tan x = - \frac{7}{24} \Rightarrow \cot x = \frac{1}{\tan} x = - \frac{24}{7}$

$\sec x = + \sqrt{{\left(- \frac{7}{24}\right)}^{2} + 1}$

$\textcolor{w h i t e}{\sec x} = \sqrt{\frac{49}{576} + 1} = \sqrt{\frac{625}{576}} = \frac{25}{24}$

$\Rightarrow \cos x = \frac{1}{\sec} x = \frac{24}{25}$

$\Rightarrow \sin x = - \sqrt{1 - {\left(\frac{24}{25}\right)}^{2}}$

$\textcolor{w h i t e}{\Rightarrow \sin x} = - \sqrt{1 - \frac{576}{625}} = - \sqrt{\frac{49}{625}} = - \frac{7}{25}$

$\Rightarrow \csc x = \frac{1}{\sin} x = - \frac{25}{7}$

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