Use the Following Information to Express the Remaining Five Trigonometric $u$

question 24

Multiple Choice

Use the following information to express the remaining five trigonometric values as functions of $u$ . Assume that $u$ is positive. Rationalize any denominators that contain radicals. $\sin \theta = - u ^ { 2 } , 270 ^ { \circ } < \theta < 360 ^ { \circ }$

• A) \(\begin{array} { l }
• \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = - \frac { 1 } { u ^ { 2 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\
• \cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } .
• \end{array}\)
• B) \(\begin{array} { l }
• \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 + u ^ { 4 } } , \\
• \cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { 1 } { u ^ { 2 } } .
• \end{array}\)
• C) \(\begin{array} { l }
• \tan \theta = - \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\
• \cot \theta = - \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = - \frac { 1 } { u ^ { 2 } } .
• \end{array}\)
• D) \(\begin{array} { l }
• \tan \theta = \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = - \sqrt { 1 - u ^ { 4 } } , \\
• \cot \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , \quad \csc \theta = \frac { 1 } { u ^ { 2 } } .
• \end{array}\)
• E) \(\begin{array} { l l }
• \tan \theta = \frac { u ^ { 2 } \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , & \sec \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { 1 - u ^ { 4 } } , \quad \cos \theta = \sqrt { 1 - u ^ { 4 } } , \\
• \cot \theta = \frac { \sqrt { 1 - u ^ { 4 } } } { u ^ { 2 } } , & \csc \theta = \frac { 1 } { u ^ { 2 } } .
• \end{array}\)

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Definitions:

Service Rate

The speed at which a service provider can deliver its service to customers.

Waiting Line

Refers to the process where customers or items wait in line for service or processing, commonly studied to improve efficiency in operations management.

Negative Exponential Distribution

A continuous probability distribution concerning time between events in a Poisson point process, representing the likelihood of wait times or life spans.

Service Time

The amount of time taken to complete a service for a customer, from initiation to completion.