Question

How do you find the value of \[\cot {300^ \circ }\] ?

Answer 1

Hint: Here in the question to find the value of \[\cot {300^ \circ }\] by using the addition and subtraction with different standard angles. Then by applying ASTC rule of trigonometry reduce the given angle \[{300^ \circ }\] to the standard angle then by that value of standard angle of trigonometric function we get the required value.

Complete step-by-step answer:
To find the exact value of \[\cot {300^ \circ }\] by using the ASTC rule of trigonometry ASTC rule stands for the "all sine tangent cosine" rule. It is intended to remind us that all trigonometric ratios are positive in the first quadrant of a graph, only the sine and its cofunction cosecant are positive in the second quadrant, only the tangent and its cofunction cotangent are positive in the third quadrant, and only the cosine and its cofunction secant are positive in the fourth quadrant. One way to remember this arrangement is with a sentence “All students take coffee” or “All science teachers are crazy”.
Then always remember, when you write the trigonometric function with angle \[{90^ \circ }\] or \[{270^ \circ }\] , the function will change to its cofunction.
Let’s \[\cot {300^ \circ }\] can be written as:
 \[\cot {300^ \circ } = \cot {\left( {270 + 30} \right)^ \circ }\]
And
 \[\cot {300^ \circ } = \cot {\left( {360 - 60} \right)^ \circ }\]
Let’s solve the \[\cot {300^ \circ } = \cot {\left( {270 + 30} \right)^ \circ }\]
The angle \[{300^ \circ }\] is greater than \[{270^ \circ }\] and thus lies in the fourth quadrant. tan and cot functions are not positive here, i.e., they are negative. In angle \[{270^ \circ }\] the function will change to its cofunction i.e., change cot to tan.
 \[
   \Rightarrow \,\,\,\cot {300^ \circ } = \cot {\left( {270 + 30} \right)^ \circ } \\
   \Rightarrow \,\,\,\cot {300^ \circ } = - \tan {30^ \circ } \;
 \]
The value of \[\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}\]
 \[\therefore \,\,\,\cot {300^ \circ } = - \dfrac{1}{{\sqrt 3 }}\]
Now, Let’s solve the \[\cot {300^ \circ } = \cot {\left( {360 - 60} \right)^ \circ }\]
The angle \[{300^ \circ }\] is lesser than \[{360^ \circ }\] and thus lies in the fourth quadrant. tan and cot functions are not positive here, i.e., they are negative. Here we must keep the function as cot itself.
 \[
   \Rightarrow \,\,\,\cot {300^ \circ } = \cot {\left( {360 - 60} \right)^ \circ } \\
   \Rightarrow \,\,\,\cot {300^ \circ } = - \cot {60^ \circ } \;
 \]
The value \[\cot {60^ \circ } = \dfrac{1}{{\sqrt 3 }}\]
 \[\therefore \,\,\,\cot {300^ \circ } = - \dfrac{1}{{\sqrt 3 }}\]
Hence, the exact value of \[\cot {300^ \circ }\] is \[ - \dfrac{1}{{\sqrt 3 }}\]
So, the correct answer is “ \[ - \dfrac{1}{{\sqrt 3 }}\] ”.

Note: The trigonometry has trigonometry ratios, sine, cosine, tangent, cosecant, secant and cotangent are the trigonometry ratios. To solve these kinds of problems we must know about the table of trigonometry ratios for standard angles. we can determine the values for the angles.