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What is the exact value of [tex]\cot \left(\frac{3 \pi}{4}\right)[/tex]?

A. [tex]-1[/tex]
B. [tex]-\sqrt{2}[/tex]
C. [tex]1[/tex]
D. [tex]\sqrt{2}[/tex]


Sagot :

To find the exact value of [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex], we start by recalling the definition and properties of the cotangent function. Cotangent, written as [tex]\(\cot(\theta)\)[/tex], is the reciprocal of the tangent function:

[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} \][/tex]

Thus, to find [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex], we first need to determine the value of [tex]\(\tan \left(\frac{3\pi}{4}\right)\)[/tex].

The angle [tex]\(\frac{3\pi}{4}\)[/tex] is in the second quadrant of the unit circle. We know that in the second quadrant, the tangent function is negative. Specifically, [tex]\(\frac{3\pi}{4}\)[/tex] relates to the angle [tex]\(\frac{\pi}{4}\)[/tex], where tangent has the value of 1, but taking into account the sign in the second quadrant, we have:

[tex]\[ \tan \left(\frac{3\pi}{4}\right) = -\tan \left(\frac{\pi}{4}\right) = -1 \][/tex]

Now, using the reciprocal relation of cotangent, we find:

[tex]\[ \cot \left(\frac{3\pi}{4}\right) = \frac{1}{\tan \left(\frac{3\pi}{4}\right)} = \frac{1}{-1} = -1 \][/tex]

Thus, the exact value of [tex]\(\cot \left(\frac{3\pi}{4}\right)\)[/tex] is:

[tex]\[ -1 \][/tex]

So, the correct answer is:

[tex]\[ -1 \][/tex]
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