Question

How do you evaluate $ \cot {{300}^{\circ }} $ ?

Answer 1

Hint: We can calculate the value of $ \cot {{300}^{\circ }} $ by using the formula $ \cot \left( {{360}^{\circ }}-\theta \right)=-\cot \theta $ we can write $ {{300}^{\circ }} $ as $ {{360}^{\circ }}-{{60}^{\circ }} $ and we know the value of $ \cot {{60}^{\circ }} $ , thus we can easily evaluate the value of $ \cot {{300}^{\circ }} $ .

Complete step by step answer:
We have to evaluate the value of $ \cot {{300}^{\circ }} $ we know the property of cot x that $ \cot \left( {{360}^{\circ }}-x \right)=-\cot x $ we can write $ {{300}^{\circ }} $ as $ {{360}^{\circ }}-{{60}^{\circ }} $
 $ \cot {{300}^{\circ }}=\cot \left( {{360}^{\circ }}-{{60}^{\circ }} \right) $
Applying the formula $ \cot \left( {{360}^{\circ }}-x \right)=-\cot x $ to the above equation
 $ \cot {{300}^{\circ }}=-\cot {{60}^{\circ }} $
We know the value of $ \cot {{60}^{\circ }} $ that is equal to $ \dfrac{1}{\tan {{60}^{\circ }}}=\dfrac{1}{\sqrt{3}} $
 $ \cot {{300}^{\circ }} $ is equal to $ -\cot {{60}^{\circ }} $ which is $ -\dfrac{1}{\sqrt{3}} $
So the value of $ \cot {{300}^{\circ }} $ is equal to $ -\dfrac{1}{\sqrt{3}} $ .

Note:
We can evaluate $ \cot {{300}^{\circ }} $ by many more method we can write $ \cot {{300}^{\circ }} $ as $ \cot \left( {{270}^{\circ }}+{{30}^{\circ }} \right) $ and we know the formula for $ \cot \left( {{270}^{\circ }}+\theta \right) $ which is equal to $ -\tan \theta $ and we know the value of $ \tan {{30}^{\circ }} $ . Another method is by using $ \cot 2x $ formula we can put 150 degrees in the place of x and find the value of $ \cot {{300}^{\circ }} $ .We can do this problem by many method but always remember the sign convention of the formula, cot x is positive when x lies in first or third quadrant and cot x is negative when x lies in second or fourth quadrant. Now we can see that 300 degrees lie in the fourth quadrant and our answer is $ -\dfrac{1}{\sqrt{3}} $ which is a negative number.
cot x is not defined when $ x=n\pi $ where n is an integer so cot x is not defined at $ -\pi $ , 0, $ \pi $ , $ 2\pi $ etc. cot x and tan x are reciprocal of each other so cot x is not defined when tan x is 0 and tan x is defined when cot x is 0 . So tan x is not defined when $ x=\dfrac{n\pi }{2} $ where n is an integer.