Answer:
The exact value is [tex]\cfrac{\sqrt{3}}3[/tex]
Step-by-step explanation:
Since 60 degrees is an angle we can find on the unit circle, the goal to get an exact value is to use the elements of the unit circle, which are exact values of sine and cosine.
Writing cotangent in terms of sine and cosine
We can use the trigonometric identity
[tex]\cot \theta = \cfrac{\cos \theta }{\sin \theta }[/tex]
Thus for the exercise we will have
[tex]\cot 60^\circ = \cfrac{\cos 60^\circ }{\sin 60^\circ }[/tex]
Identifying the known exact values.
From the unit circle that you can see on the attached image below, we have to identify the exact values of cosine and sine of 60 degrees.
So first try to look for the angle 60 degrees, there you will see a point that has a pair of values, those represent (cosine, sine), thus we get:
[tex]\cos 60^\circ=\cfrac 12 \\\\\sin 60^\circ = \cfrac{\sqrt3}2[/tex]
Finding the exact value of cot 60 degrees.
We can replace the exact values of sine and cosine on the trigonometric identity for cotangent.
[tex]\cot 60^\circ = \cfrac{\cfrac 12 }{\cfrac{\sqrt 3}2 }[/tex]
Working with the reciprocal we get
[tex]\cot 60^\circ = \cfrac 12\times \cfrac2{\sqrt 3}[/tex]
Simplifying we get
[tex]\cot 60^\circ = \cfrac 1{\sqrt 3}[/tex]
Rationalizing since we usually do not want square roots on the denominator we get
[tex]\cot 60^\circ = \cfrac 1{\sqrt 3} \times \cfrac{\sqrt 3}{\sqrt 3}\\\boxed{\cot 60^\circ = \cfrac {\sqrt 3}3}[/tex]
And that is the exact value of cotangent of 60 degrees.
