Question

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

Answer 1

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.