contestada

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

Respuesta :

jacob193

Answer:

The angle is in the second quadrant.

Step-by-step explanation:

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

[tex]\displaystyle \csc t = \frac{1}{\sin t}[/tex].

Therefore, [tex]\csc(t) > 0[/tex] is equivalent to [tex]\sin (t) > 0[/tex].

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

  • [tex]\cos (t) = x[/tex], and
  • [tex]\sin(t) = y[/tex].

For angle [tex]t[/tex],

  • [tex]x = \cos(t) < 0[/tex], meaning that the intersection is to the left of the [tex]y[/tex]-axis.
  • [tex]y = \sin(t) > 0[/tex], meaning that the intersection is above the [tex]x[/tex]-axis.

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

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