At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Our community provides timely and precise responses to help you understand and solve any issue you face.
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]
[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]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.