How do you use $csc θ = 5$ $csctheta=5$ to find $cot θ$ $cottheta$ ?

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How do you use $csc θ = 5$ $csctheta=5$ to find $cot θ$ $cottheta$ ?

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Answer 1 · 2017-03-27T01:20:05

$cot θ = \pm 2 \sqrt{6}$ $cottheta=+-2sqrt6$

Explanation:

The terminal ray of an angle $θ$ $theta$ in standard position intersects a point $(x, y)$ $(x,y)$ on the unit circle such that $csc θ = \frac{1}{y}$ $csctheta=1/y$ , $cot θ = \frac{x}{y}$ $cottheta=x/y$ , and $x^{2} + y^{2} = 1$ $x^2+y^2=1$ .

Knowing that $csc θ = 5$ $csctheta=5$ we know that $y = \frac{1}{5}$ $y=1/5$

Since $x^{2} + y^{2} = 1$ $x^2+y^2=1$ , $x = \pm \sqrt{1 - y^{2}}$ $x=+-sqrt(1-y^2)$

$cot θ = \frac{x}{y} = \frac{\pm \sqrt{1 - y^{2}}}{y} = \frac{\pm \sqrt{1 - {(\frac{1}{5})}^{2}}}{\frac{1}{5}}$ $cottheta=x/y=(+-sqrt(1-y^2))/y=(+-sqrt(1-(1/5)^2))/(1/5)$
$cottheta=+-5sqrt(1-1/25)=+-5sqrt(24/25)=+-cancel5sqrt24/cancel5=+-2sqrt6$

Answer 2 · 2017-03-27T04:18:08

$+- 2sqrt6$

Explanation:

Use trig identity:
$1 + cot^2 t = csc^2 t$
In this case:
$1 + cot^2 t = 25$
$cot^2 t = 24$
$cot t = +- 2sqrt6$

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How do you use $csc θ = 5$ $csctheta=5$ to find $cot θ$ $cottheta$ ?

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