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.`