Question

How do you evaluate the expression \[\cot \left( {{{45}^ \circ }} \right)\]?

Answer 1

Hint: Here the question is related to trigonometry, we use the trigonometry ratios and we are going to solve this question. By using the trigonometry properties we are going to solve this problem. To find the value we need the table of trigonometry ratios for standard angles.

Complete step by step solution:
The question is related to trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tan, cosec, sec, and cot.
The cosecant, secant and cotangent trigonometric ratios are the reciprocal of sine, cosine and tangent trigonometric ratio respectively.
Now consider the given question
 \[\cot \left( {{{45}^ \circ }} \right)\]
This will lie in the first quadrant. In trigonometry we have ASTC rules for the trigonometry ratios. The above inequality lies in the first quadrant and the cotangent trigonometry ratio is positive in the first quadrant.
Now we consider the table
So we have a table for the trigonometry ratio sine for the standard angles.

Angle	0	30	45	60	90
cotangent	\[\infty \]	\[\sqrt 3 \]	\[1\]	\[\dfrac{1}{{\sqrt 3 }}\]	0

The above table is given for the standard degree. The given question is in the form of degree
Therefore the value of \[\cot ({45^ \circ })\] is \[1\].

Note:
 In trigonometry to find the value of angles we have a table of trigonometry ratios for the standard angles. The ASTC rule is applicable for the highest values. Whether the value of angle is in degree or radians the value for the standard angles will not change. Where ASTC rule is abbreviated as ALL SINE TANGENT COSINE.