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

question 6

Multiple Choice

Use the following information to express the remaining five trigonometric values as functions of $t$ . Assume that $t$ is positive. Rationalize any denominators that contain radicals. $\cos \theta = - \frac { 3 t } { 4 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }$

• A) \(\begin{array} { l }
• \tan \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = - \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\
• \cot \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .
• \end{array}\)
• B) \(\begin{array} { l }
• \tan \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \sec \theta = - \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\
• \cot \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .
• \end{array}\)
• C) \[\begin{array} { l }
• \tan \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 } { 3 t } , \quad \sin \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\
• \cot \theta = \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = - \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .
• \end{array}\]
• D) \(\begin{array} { l }
• \tan \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 } { 3 t } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\
• \cot \theta = \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } .
• \end{array}\)
• E) \(\begin{array} { l }
• \tan \theta = - \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 3 t } , \quad \sec \theta = \frac { 4 \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \sin \theta = \frac { \sqrt { 16 - 9 t ^ { 2 } } } { 4 } , \\
• \cot \theta = - \frac { 3 t \sqrt { 16 - 9 t ^ { 2 } } } { 16 - 9 t ^ { 2 } } , \quad \csc \theta = - \frac { 4 } { 3 t } .
• \end{array}\)

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Definitions:

Objectivity

The quality of being unbiased, impartial, or not influenced by personal feelings or opinions in considering and representing facts.

Self-actualization

The realization or fulfillment of one's talents and potentialities, often considered as the motive to achieve the highest personal potential in Maslow's hierarchy of needs.

Physiological Needs

Fundamental requirements for human survival, such as air, water, food, shelter, and sleep, forming the base level of Maslow's hierarchy of needs.

Spiritual Development

The process of evolving one's personal sense of meaning and purpose, often through connection with something greater than oneself.