How do you calculate ${cos}^{- 1} (- 0.8)$ $cos^-1 (-0.8)$ ?

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How do you calculate ${cos}^{- 1} (- 0.8)$ $cos^-1 (-0.8)$ ?

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Answer 1 · 2015-12-16T02:10:24

$θ = {143.13}^{\circ}$ $theta=143.13^@$ or ${216.87}^{\circ}$ $216.87^@$

Explanation:

Since -0.8 is not a value you would find in a trigonometric ratio, which are: $sin θ$ $sintheta$ , $cos θ$ $costheta$ , $tan θ$ $tantheta$ , $csc θ$ $csctheta$ , $sec θ$ $sectheta$ , $cot θ$ $cottheta$ , you have to use a calculator to find the answer:

$cos θ = - 0.8$ $costheta=-0.8$
$\times θ = {cos}^{- 1} (- 0.8)$ $color(white)(xx)theta=cos^-1(-0.8)$
$\times x \approx {143.13}^{\circ}$ $color(white)(xxx)~~143.13^@$

However, according to the CAST rule, cosine is also negative in the third quadrant, which means there is a second possible answer.

![http://mathonline.wikidot.com/cast-rule]( useruploads.socratic.org )

First, find the related acute angle, and add the angle to $180^{\circ}$ $180^@$ to find the second possible answer.

Finding the related acute angle
$180^{\circ} - {143.13}^{\circ}$ $180^@-143.13^@$
$= {36.87}^{\circ}$ $=36.87^@$

Finding the second possible angle
$= 180^{\circ} + {36.87}^{\circ}$ $=180^@+36.87^@$
$= {216.87}^{\circ}$ $=216.87^@$

$:.$ , the two possible angles are $143.13^@$ and $216.87^@$ .

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