Let $\theta$ Be An Angle In Standard Position. Name The Quadrant In Which $\theta$ Lies Given:$\tan \theta \ \textless \ 0$, $\csc \theta \ \textless \ 0$.The Angle $\theta$ Lies In Which Quadrant?A. III

$$

Introduction

In trigonometry, angles in standard position are a crucial concept to understand. An angle in standard position is formed by a point on the unit circle and the positive x-axis. The quadrant in which an angle lies is determined by the signs of the trigonometric functions associated with that angle. In this article, we will explore how to determine the quadrant in which an angle $\theta$ lies given that $\tan \theta < 0$ and $\csc \theta < 0$.

Understanding the Trigonometric Functions

Before we proceed, let's briefly review the trigonometric functions and their signs in different quadrants.

Function	Quadrant I	Quadrant II	Quadrant III	Quadrant IV
$\sin \theta$	+	+	-	-
$\cos \theta$	+	-	-	+
$\tan \theta$	+	-	+	-
$\csc \theta$	+	-	-	+
$\sec \theta$	+	-	+	-
$\cot \theta$	+	+	-	-

Given Conditions

We are given that $\tan \theta < 0$ and $\csc \theta < 0$. Let's analyze these conditions to determine the quadrant in which $\theta$ lies.

$\tan \theta < 0$

The tangent function is negative in Quadrants II and IV. This means that $\theta$ lies in either Quadrant II or Quadrant IV.

$\csc \theta < 0$

The cosecant function is negative in Quadrants III and IV. This means that $\theta$ lies in either Quadrant III or Quadrant IV.

Combining the Conditions

Now, let's combine the two conditions to determine the quadrant in which $\theta$ lies.

Since $\tan \theta < 0$, we know that $\theta$ lies in either Quadrant II or Quadrant IV.

Since $\csc \theta < 0$, we know that $\theta$ lies in either Quadrant III or Quadrant IV.

Combining these two conditions, we can conclude that $\theta$ lies in Quadrant IV.

Conclusion

In conclusion, given that $\tan \theta < 0$ and $\csc \theta < 0$, the angle $\theta$ lies in Quadrant IV.

Example

Let's consider an example to illustrate this concept.

Suppose we have an angle $\theta$ such that $\tan \theta = -1$ and $\csc \theta = -2$. We can use the unit circle to visualize this angle.

The angle $\theta$ lies in Quadrant IV, as the tangent function is negative and the cosecant function is negative in this quadrant.

Final Thoughts

$$$$$$

Q: What is the standard position of an angle?

A: An angle in standard position is formed by a point on the unit circle and the positive x-axis. The vertex of the angle is at the origin (0, 0) of the coordinate plane.

Q: How do I determine the quadrant of an angle using the unit circle?

A: To determine the quadrant of an angle using the unit circle, you can use the following steps:

1. Draw the unit circle and label the x and y axes.
2. Draw the terminal side of the angle, which is the line segment that extends from the origin to the point on the unit circle.
3. Determine the quadrant in which the terminal side of the angle lies by looking at the signs of the x and y coordinates.

Q: What is the relationship between the trigonometric functions and the quadrants?

A: The trigonometric functions have the following relationships with the quadrants:

Function	Quadrant I	Quadrant II	Quadrant III	Quadrant IV
$\sin \theta$	+	+	-	-
$\cos \theta$	+	-	-	+
$\tan \theta$	+	-	+	-
$\csc \theta$	+	-	-	+
$\sec \theta$	+	-	+	-
$\cot \theta$	+	+	-	-

Q: How do I determine the quadrant of an angle using the trigonometric functions?

A: To determine the quadrant of an angle using the trigonometric functions, you can use the following steps:

1. Evaluate the trigonometric functions of the angle.
2. Use the signs of the trigonometric functions to determine the quadrant in which the angle lies.

Q: What is the relationship between the tangent and cosecant functions and the quadrants?

A: The tangent and cosecant functions have the following relationships with the quadrants:

• The tangent function is negative in Quadrants II and IV.
• The cosecant function is negative in Quadrants III and IV.

Q: How do I determine the quadrant of an angle using the tangent and cosecant functions?

A: To determine the quadrant of an angle using the tangent and cosecant functions, you can use the following steps:

1. Evaluate the tangent and cosecant functions of the angle.
2. Use the signs of the tangent and cosecant functions to determine the quadrant in which the angle lies.

Q: What is the final answer to the problem of determining the quadrant of an angle given that $\tan \theta < 0$ and $\csc \theta < 0$?

A: The final answer to the problem of determining the quadrant of an angle given that $\tan \theta < 0$ and $\csc \theta < 0$ is that the angle $\theta$ lies in Quadrant IV.

Q: Can you provide an example to illustrate this concept?

A: Yes, let's consider an example to illustrate this concept.

Suppose we have an angle $\theta$ such that $\tan \theta = -1$ and $\csc \theta = -2$. We can use the unit circle to visualize this angle.

The angle $\theta$ lies in Quadrant IV, as the tangent function is negative and the cosecant function is negative in this quadrant.

Q: What are some common mistakes to avoid when determining the quadrant of an angle?

A: Some common mistakes to avoid when determining the quadrant of an angle include:

• Not using the correct signs of the trigonometric functions.
• Not using the correct quadrant for the angle.
• Not considering the relationship between the tangent and cosecant functions and the quadrants.

Q: How can I practice determining the quadrant of an angle?

A: You can practice determining the quadrant of an angle by:

• Using the unit circle to visualize the angle.
• Evaluating the trigonometric functions of the angle.
• Using the signs of the trigonometric functions to determine the quadrant in which the angle lies.
• Practicing with different angles and trigonometric functions.