# How do you find the value of cot(arccos (-2/3))?

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You might know that you can use the formula cot (x) = cos(x) / sin(x) to convert between the trig functions, but how do you find the value of that exact fraction?

There’s an easy way to approach this problem using a process called substitution.

With just a few steps, you can figure out how to find the value of cot(arccos (-2/3)). All you need are your calculator and pencil and paper!

## Explaination 1

Suppose that a certain acute angle u satisfies

cosu=2/3

Then the Pythagorean theorem shows that

sinu=5–√/3

So

cotu=5–√/2

So

cotarccos(−2/3)=−cotu=−5–√/2

## Explaination 2

How do you find the value of cot(arccos (-2/3))?
Suppose that a certain acute angle u satisfies

cosu=2/3

Then the Pythagorean theorem shows that

sinu=5–√/3

So

cotu=5–√/2

So

cotarcs (−2 / 3)=−cotu=−5–√/2

The same way that you evaluate any trig-of-inverse-trig expression:

Draw a right triangle in the appropriate quadrant that shows an angle giving the trig value that you want (for arcsin and arctan, first or fourth; for arccos, first or second).

Fill in the missing side using the Pythagorean Theorem (it will always be positive, due to our choice of quadrant).

Read off the value of the outside trig function from the triangle.
In this case:

Your triangle would have adjacent side -2 (2 units leftward) and hypotenuse 3, by soh-cah-toa.

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The opposite leg would be 32−22−−−−−−√=5–√ (upward).

The cotangent is adjacent over opposite, so you’d get −25√=−25√5 .

## Explaination 3

We know that arccos(-2/3) is an angle since we are calculating its cotangent.

Also if there are 2 possible answers for arccos(-2/3), then there must be two answers for its cotangent. See diagram below: Notice that 180 +/- theta have arccos(-2/3). In other words, cos of either angle gives you -2/3.

Since the 3rd side must be sqroot5, we can calculate cotangents of these angles as well. Cot theta is -2/sqroot5 or -2/-sqroot5.

So final answer assuming range of 0 to 2pi is +/- (2sqroot5)/5. 