Question

Find the value of \[\cot ( - {315^ \circ })\].

Answer 1

Hint: The calculation of lengths and angles of the right-angle triangle is dealt with by trigonometry values of various ratios, such as sine, cosine, tangent, secant, cotangent, and cosecant. So to solve trigonometry problems, the values of trigonometric functions for \[{0^ \circ }\],\[{30^ \circ }\],\[{45^ \circ }\],\[{60^ \circ }\], and \[{90^ \circ }\] are widely used.

Complete step by step solution:
We have to find out the value of \[\cot ( - {315^ \circ })\].

Find the angle in the first quadrant with equal trig values to use as a reference angle. Since cotangent is negative in the fourth quadrant, make the term negative. The fundamental method for determining the related angle for a given angle is to subtract \[{180^ \circ }\] from \[\theta \] or to subtract \[\theta \] from \[{180^ \circ }\] or \[{360^ \circ }\].
We have,
\[\cot ( - \theta ) = - \cot (\theta )\]
Therefore,
\[\cot ( - {315^ \circ }) = - \cot ({315^ \circ })\]
\[ = - \cot ({360^ \circ } - {45^ \circ })\]
We know that,
\[\cot (2\prod - \theta ) = - \cot \theta \]
Therefore,
\[\cot ({360^ \circ } - {45^ \circ }) = - ( - \cot {45^ \circ })\]= $\cot 45^\circ$
We also know that,
\[\cot {45^ \circ } = 1\]
Hence,
\[\cot ( - {315^ \circ }) = 1\]
Therefore, the value of \[\cot ( - {315^ \circ })\] is $1$.

Note:
> We can conclude that, to find the trigonometric ratio of an angle between \[{0^ \circ }\] and \[{360^ \circ }\], we should follow these steps:
> First we should change the given angle to a related angle using equations. Then obtain the ratio's sign by noting the quadrant, then evaluate the corresponding angle's trigonometric ratio and assign the required sign.
> \[\cot ( - 315)\] Indicates that the value at angle x is the inverse of the value at angle -x. To put it another way,
\[\cot (x) = - \cot (x)\]. With a \[{360^ \circ }\] span, the cotangent function is periodic. This property indicates that the values of the function repeat every \[{360^ \circ }\]. The cotangent is the inverse of the tangent in trigonometry. The tangent is calculated by dividing the opposite side of a triangle by the adjacent side.