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Use the given information to determine the exact trigonometric value.

Given: [tex]\(\cot \theta = -\frac{\sqrt{5}}{2}, \frac{\pi}{2} \ \textless \ \theta \ \textless \ \pi\)[/tex]

Find: [tex]\(\tan \theta\)[/tex]

A. [tex]\(-\frac{\sqrt{5}}{5}\)[/tex]

B. [tex]\(-\frac{2 \sqrt{5}}{5}\)[/tex]

C. [tex]\(-\frac{\sqrt{5}}{2}\)[/tex]

D. [tex]\(-\frac{2 \sqrt{5}}{2}\)[/tex]


Sagot :

To determine the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(\cot \theta = -\frac{\sqrt{5}}{2}\)[/tex] and [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], we can follow these steps:

1. Understand the given information:

[tex]\(\cot \theta = -\frac{\sqrt{5}}{2}\)[/tex]

The cotangent function [tex]\(\cot \theta\)[/tex] is the reciprocal of the tangent function [tex]\(\tan \theta\)[/tex]. Thus, we have:

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

2. Express [tex]\(\tan \theta\)[/tex] in terms of [tex]\(\cot \theta\)[/tex]:

Given [tex]\(\cot \theta\)[/tex], we can find [tex]\(\tan \theta\)[/tex] by taking the reciprocal of [tex]\(\cot \theta\)[/tex]:

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

3. Substitute the given value of [tex]\(\cot \theta\)[/tex]:

[tex]\[ \tan \theta = \frac{1}{-\frac{\sqrt{5}}{2}} = \frac{2}{-\sqrt{5}} \][/tex]

4. Simplify the expression:

Simplify [tex]\(\tan \theta\)[/tex] by rationalizing the denominator:

[tex]\[ \tan \theta = \frac{2}{-\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \cdot (-1) = -\frac{2\sqrt{5}}{5} \][/tex]

Therefore, the correct value of [tex]\(\tan \theta\)[/tex] is [tex]\(\boxed{-\frac{2\sqrt{5}}{5}}\)[/tex], which matches option c.