Question

Let 0 be an angle in standard position. Name the quadrant in which 0 lies. sin0 <0, sec 0 >0The angle 0 lies in which quadrant?

Answer 1

The angle 0 lies in the fourth quadrant, where sin0 is negative and sec 0 is positive.

Step-by-step explanation:

Given the conditions that sin0 < 0 and sec 0 > 0, we can determine which quadrant the angle 0 lies in. The sine of an angle is negative in the third and fourth quadrants, while the secant (which is the reciprocal of the cosine) is positive in the first and fourth quadrants. Therefore, the only quadrant where both conditions are satisfied is the fourth quadrant.

Understanding the trigonometric relationships between sine and secant functions provides valuable insights into the placement of the angle within the coordinate system. With Sin x indicating the third and fourth quadrants and sec x suggesting the first and fourth quadrants, the intersection of these conditions points conclusively to the fourth quadrant as the location for the angle. This analytical approach not only clarifies the quadrant in question but also showcases the interconnected nature of trigonometric functions in determining angular positions.

Answer 2

`\sin \theta\text{ < 0 , sec}\theta\text{ >0}`

We first need to analyze them individually,

`\begin{gathered} \text{For }\sin \theta<0 \\ \text{Therefore, }\theta\text{ lies in the third and fourth quadrant } \end{gathered}`
`\begin{gathered} \text{For sec}\theta>0 \\ \text{Therefore, }\theta\text{ lies in the first and fourth quadrant} \end{gathered}`

The two conditions apply only to the fourth quadrant.

Therefore, the angle lies in the fourth quadrant IV