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

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

Explanation

Solution

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,π6,π40,\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){\cot ^{ - 1}}( - 5) , we know that cot⁡−1x=tan⁡−11x{\cot ^{ - 1}}x = {\tan ^{ - 1}}\dfrac{1}{x} , so –
cot⁡−1(−5)=tan⁡−1(−15){\cot ^{ - 1}}( - 5) = {\tan ^{ - 1}}( - \dfrac{1}{5})
We can calculate the value of tan⁡−1(−15){\tan ^{ - 1}}( - \dfrac{1}{5}) by using a calculator as –
tan⁡−1(−15)=−11.3099325∘{\tan ^{ - 1}}( - \dfrac{1}{5}) = - 11.3099325^\circ
The correct way to write −11.3099325∘- 11.3099325^\circ is 180∘−11.3099325∘=179.802604∘180^\circ - 11.3099325^\circ = 179.802604^\circ
Hence, cot⁡−1(−5){\cot ^{ - 1}}( - 5) is equal to −11.3099325∘- 11.3099325^\circ or 179.802604∘179.802604^\circ .

Note: We can find the inverse trigonometric values of the numbers associated with the basic angles like 0,π6,π4,π3,π20,\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∘- 11.3099325^\circ can be written as n180∘−11.3099325∘=179.802604∘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.