Respuesta :

Given:

[tex]\begin{gathered} csc\theta=\frac{7}{6} \\ \\ \cos \theta<0 \end{gathered}[/tex]

Let's use a trig identity to find the value of tanθ.

Apply the definition of cosecant to find known sides of the unit circle.

We have:

[tex]\csc \theta=\frac{hypotenuse}{opposite}=\frac{7}{6}[/tex]

Here, the hypotenuse and opposite sides are known.

Also, find the adjacent side using Pythagorean Theorem:

[tex]\begin{gathered} \text{adjacent}=\sqrt[]{hypotenuse^2-opposite^2} \\ \\ \text{adjacent}=\sqrt[]{7^2-6^2} \\ \\ \text{adjacent}=\sqrt[]{49-36}=\sqrt[]{13} \end{gathered}[/tex]

Apply the definition of tangent to find tanθ:

[tex]\begin{gathered} \tan \theta=\frac{opposite}{\text{adjacent}} \\ \\ \text{tan}\theta=\frac{6}{\sqrt[]{13}} \end{gathered}[/tex]

Simplify:

[tex]\begin{gathered} \tan \theta=\frac{6}{\sqrt[]{13}}\times\frac{\sqrt[]{13}}{\sqrt[]{13}} \\ \\ \tan \theta=\frac{6\sqrt[]{13}}{13}=1.664 \end{gathered}[/tex]

ANSWER:

[tex]\tan \theta=1.664[/tex]

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