Question

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

Answer 1

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 $\sin (t) \ne 0$,

$\displaystyle \csc t = \frac{1}{\sin t}$.

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.