Problem 915 Show that if \(\mathrm{x}>0\)... [FREE SOLUTION]

Chapter 28: Problem 915

Show that if \(\mathrm{x}>0\), then \(\operatorname{Arctan} \mathrm{x}=\operatorname{Arccot}(1 / \mathrm{x})\)

Short Answer

Expert verified

To show that if \(x>0\), then \(\operatorname{Arctan}(x)=\operatorname{Arccot}(1/x)\), we first recall the reciprocal relationship between tangent and cotangent functions: \(cot(\theta) = \frac{1}{tan(\theta)}\). Substituting the expressions, we get \(\operatorname{Arctan}(tan(\theta))\) and \(\operatorname{Arccot}(cot(\alpha))\). Using the relationship \(tan(\theta) = cot(\theta - \frac{\pi}{2})\), we find that \(\theta = \alpha - \frac{\pi}{2}\), which leads to the final result: \(\operatorname{Arctan}(x) = \operatorname{Arccot}(1/x)\) for \(x > 0\).

Step by step solution

Define the arctangent and arccotangent functions

The arctangent function, written as Arctan(x) or atan(x), represents the angle, in the range of (-蟺/2, 蟺/2), whose tangent equals x. Similarly, the arccotangent function, written as Arccot(x) or acot(x), represents the angle, in the range of (0, 蟺), whose cotangent equals x.

Recall the tangent and cotangent functions' relationship

The tangent and cotangent functions have a reciprocal relationship. If x > 0, we can write: \(cot(\theta) = \frac{1}{tan(\theta)}\)

Substitute the expressions

Using step 2, we can substitute the tangent of the arctangent function as follows: \(\operatorname{Arctan}(x) = \operatorname{Arctan}\left(\frac{1}{\frac{1}{tan(\theta)}}\right)\) \(\operatorname{Arctan}(x) = \operatorname{Arctan}(tan(\theta))\) And we can substitute the cotangent of the arccotangent function: \(\operatorname{Arccot}(1/x) = \operatorname{Arccot}\left(1\left(\frac{1}{cot(\alpha)}\right)\right)\) \(\operatorname{Arccot}(1/x) = \operatorname{Arccot}(cot(\alpha))\)

Show the equality

Now, we can find a relationship between 胃 and 伪. From step 3, we can say that: \(tan(\theta) = x\) \(cot(\alpha) = \frac{1}{x}\) Using the relationship from step 2, we can write the second equation as: \(tan(\theta) = \\cot(\alpha) = \\cot(\theta - \frac{\pi}{2})\) Now, we can say that: \(\theta = \alpha - \frac{\pi}{2}\) If we plug this relationship into one of the equations from step 3: \(\operatorname {Arctan}(tan(\theta)) = \operatorname {Arccot}(cot(\alpha))\) \(\operatorname {Arctan}(tan(\alpha - \frac{\pi}{2})) = \operatorname {Arccot}(cot(\alpha))\) Now, we have shown that: \(\operatorname{Arctan}(x) = \operatorname{Arccot}(1/x)\) for x > 0.

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91影视

Show that if \(\mathrm{x}>0\), then \(\operatorname{Arctan} \mathrm{x}=\operatorname{Arccot}(1 / \mathrm{x})\)

Short Answer

Step by step solution

Define the arctangent and arccotangent functions

Recall the tangent and cotangent functions' relationship

Substitute the expressions

Show the equality

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