Question

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

Accepted Answer

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.

Math

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

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