If `costheta = 2/3`  and `270^o lt thetalt 360^o` , then determine the exact value of `1/cot theta` . special triangles.

Note that a cosine function is the ratio of the side adjacent to the given angle to the hypotenuse of the triangle.

`cos theta = (adjacent side)/(hypoten u se)`  

And the given value of cos theta is:

`cos theta = 2/3`

Base on this, the length of the side adjacent to theta is 2 units and the hypotenuse is 3 units.

In order to solve the value of 1/(cot theta), the length of the side opposite to theta must be determined. To do so, use the Pythagorean formula which is:

`(opposite)^2 + (adjacent)^2 = (hypoten u se)^2`

`(opposite)^2 + 2^2 = 9`

`(opposite)^2 + 4 = 9`

`(opposite)^2 = 5`

`opposite = sqrt 5`

Hence, the length of the three sides of the triangle are:

> adjacent side of theta= 2 units,

> opposite side of theta=`sqrt5` units and

> hypotenuse =3 units

Note that the reciprocal of cotangent is tangent. So,

`1/(cot theta) = tan theta`

And tangent function is the ratio of the opposite side and adjacent side.

`tan theta = (opposite )/(adjacent)`

Substitute the length of the opposite and adjacent side of theta.

`tan theta = sqrt5/2`

Since the given interval of  `theta`  is  `270^0lt thetalt360^o`  ,  this means that angle  `theta`  lies at the fourth quadrant. And at the fourth quadrant, tangent function is negative.

Hence , `1/(cot theta) = tan theta = -sqrt5/2.`