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Determine the quadrant when the terminal side of the angle lies according to the following conditions: cos (t) < 0, csc (t) > 0.

User Picaso
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1 Answer

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Answer:

The angle is in the second quadrant.

Explanation:

The cosecant of an angle is the same as the reciprocal of the sine of that angle. In other words, as long as
\sin (t) \\e 0,


\displaystyle \csc t = (1)/(\sin t).

Therefore,
\csc(t) > 0 is equivalent to
\sin (t) > 0.

Consider a unit circle centered at the origin. If the terminal side of angle
t intersects the unit circle at point
(x,\, y), then


  • \cos (t) = x, and

  • \sin(t) = y.

For angle
t,


  • x = \cos(t) < 0, meaning that the intersection is to the left of the
    y-axis.

  • y = \sin(t) > 0, meaning that the intersection is above the
    x-axis.

In other words, this intersection is above and to the left of the origin. That corresponds to second quadrant of the cartesian plane.

User IgorCh
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6.4k points