Inverse cotangent function theory and graph

The cotangent function can be written as cotθ = x. Taking the inverse on each side, we get the required inverse cotangent function i.e. cot-1x = θ. It can also be written as arccot(x) = θ. The concept of arccot comes from the cotangent of a arc drawn on a circumcircle. We also know that tangent of an angle is opposite side divided by the adjacent one. The cotangent angle is just opposite to the tangent i.e. adjacent side divided by opposite. This yields us one more formula for inverse cotangent i.e. cot-1(adjacent / opposite) = θ. To elaborate more on the topic, lets move on to the graph of inverse cot function and list out its important features.

Graph of inverse cot function & important features

As shown in below figure, the graph of inverse cotangent extends in both negative as well as positive directions.
graph of inverse cotangent

The key features are listed below which describes the inverse tangent function in the perfect way.

  • The domain of inverse cotangent function is in between (-∞, +∞).

  • The range of the inverse cot function is either (-π/2, 0) or (0, π/2).

  • cot(cot-1y) = y.

  • cot-1y = π/2 – tan-1y for any values of y.

Now that we have discussed the key features of inverse cotangent along with its graph, its time to go for problems and check out your knowledge.

(1) Find the value of arccot(1) using inverse tangent function:

We have arccot(y) = π/2 – arctan(y). Hence arccot(1) = π/2 – arctan(1). Since tan(π/4) = 1, we have arctan(1) = π/4. Thus arccot(1) = π/2 – π/4 = π/4 rad. Hence the value of arccot(1) is π/4 radians or 45 degrees.

(2) Find cotangent angle if length of adjacent side is 3 cm and that of opposite side is 4 cm:

The inverse cotangent function gives the cotangent angle. In this way, cotangent angle θ can be given by θ = arccot(adjacent / opposite) = arccot(3/4) = 90 – arctan(3/4) = 53.14 degrees.

Thus we had a detailed theory on inverse cotangent functions along with graph,features and problems. The inverse cotangent function like other inverse trig functions finds several applications in the fields of technology and science.

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