Question

How do you find the exact value of ${\cot ^{ - 1}}( - 5)$ ?

Answer 1

Hint: To observe the relation between the sides and the right-angled triangle we use trigonometry functions. The trigonometric ratios are simply the ratio of two sides of a right-angled triangle. The inverse of the trigonometric functions is known as the inverse trigonometric functions, that is, we know the numerical value and we have to find the angle associated with it. We know the cotangent value of some basic angles like $0,\dfrac{\pi }{6},\dfrac{\pi }{4}$ etc. So finding out the angle whose cotangent is equal to -5, we can find the correct answer.

Complete step-by-step solution:
We have to find out ${\cot ^{ - 1}}( - 5)$ , we know that ${\cot ^{ - 1}}x = {\tan ^{ - 1}}\dfrac{1}{x}$ , so –
${\cot ^{ - 1}}( - 5) = {\tan ^{ - 1}}( - \dfrac{1}{5})$
We can calculate the value of ${\tan ^{ - 1}}( - \dfrac{1}{5})$ by using a calculator as –
${\tan ^{ - 1}}( - \dfrac{1}{5}) = - 11.3099325^\circ $
The correct way to write $ - 11.3099325^\circ $ is $180^\circ - 11.3099325^\circ = 179.802604^\circ $
Hence, ${\cot ^{ - 1}}( - 5)$ is equal to $ - 11.3099325^\circ $ or $179.802604^\circ $ .

Note: We can find the inverse trigonometric values of the numbers associated with the basic angles like $0,\dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{\pi }{3},\dfrac{\pi }{2}$ and the angles related to them but the given value cannot be found by any identity or formula that’s why we have to use the calculator. The signs of the trigonometric functions are different in different quadrants.

Values of all the trigonometric functions are positive in the first quadrant; in the second quadrant, only sine functions have positive value; tangent functions are positive in the third quadrant and in the fourth quadrant only cosine functions are positive. There can be infinite answers to the given question, as $ - 11.3099325^\circ $ can be written as $n180^\circ - 11.3099325^\circ = 179.802604^\circ $ where n can take a large variety of values, but in this question, we consider only one value.