Recalling the Definition of the Cosecant Function

Oct 13, 2025 by windailys.com 104 views

Understanding the Problem

The problem requires us to draw a horizontal, vertical, or diagonal line to represent the equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ and use the line to help us solve the equation for $0 \leq \theta \ \textless \ 2 \pi$ . We will express our answer in both radians and degrees.

Recalling the Definition of the Cosecant Function

The cosecant function is defined as the reciprocal of the sine function. In other words, $\csc \theta = \frac{1}{\sin \theta}$ . This means that the equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ is equivalent to $\frac{1}{\sin \theta} = -\frac{2 \sqrt{3}}{3}$ .

Drawing a Line Representation

To draw a line representation of the equation, we need to find the value of $\sin \theta$ that satisfies the equation. We can do this by taking the reciprocal of both sides of the equation: $\sin \theta = -\frac{3}{2 \sqrt{3}}$ . Simplifying this expression, we get $\sin \theta = -\frac{\sqrt{3}}{2}$ .

Finding the Reference Angle

The sine function is negative in the third and fourth quadrants. To find the reference angle, we need to find the angle in the first quadrant that has the same absolute value of sine as the given angle. The reference angle for $\sin \theta = -\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ .

Finding the Angles in the Third and Fourth Quadrants

Since the sine function is negative in the third and fourth quadrants, we need to add $\pi$ to the reference angle to find the angles in these quadrants. The angles in the third and fourth quadrants are $\frac{\pi}{3} + \pi = \frac{4\pi}{3}$ and $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$ , respectively.

Expressing the Answer in Both Radians and Degrees

We have found the angles in the third and fourth quadrants that satisfy the equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ . To express the answer in both radians and degrees, we need to convert the angles from radians to degrees. Using the conversion factor $\pi \text{ radians} = 180^\circ$ , we get:

$\frac{4\pi}{3} \text{ radians} = \frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ$
$\frac{2\pi}{3} \text{ radians} = \frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ$

Conclusion

In this article, we have solved the equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ using a line representation. We have found the angles in the third and fourth quadrants that satisfy the equation and expressed the answer in both radians and degrees. The angles in the third and fourth quadrants are $\frac{4\pi}{3}$ and $\frac{2\pi}{3}$ , respectively, and in degrees, they are $240^\circ$ and $120^\circ$ , respectively.

Key Takeaways

The cosecant function is defined as the reciprocal of the sine function.
The equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ is equivalent to $\frac{1}{\sin \theta} = -\frac{2 \sqrt{3}}{3}$ .
The sine function is negative in the third and fourth quadrants.
The reference angle for $\sin \theta = -\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ .
The angles in the third and fourth quadrants are $\frac{4\pi}{3}$ and $\frac{2\pi}{3}$ , respectively.

Final Answer

The final answer is $\boxed{\frac{4\pi}{3}, \frac{2\pi}{3}}$ .

Understanding the Problem

Q1: What is the definition of the cosecant function?

A1: The cosecant function is defined as the reciprocal of the sine function. In other words, $\csc \theta = \frac{1}{\sin \theta}$ .

Q2: How do we draw a line representation of the equation?

A2: To draw a line representation of the equation, we need to find the value of $\sin \theta$ that satisfies the equation. We can do this by taking the reciprocal of both sides of the equation: $\sin \theta = -\frac{3}{2 \sqrt{3}}$ . Simplifying this expression, we get $\sin \theta = -\frac{\sqrt{3}}{2}$ .

Q3: What is the reference angle for $\sin \theta = -\frac{\sqrt{3}}{2}$ ?

A3: The reference angle for $\sin \theta = -\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ .

Q4: How do we find the angles in the third and fourth quadrants?

A4: Since the sine function is negative in the third and fourth quadrants, we need to add $\pi$ to the reference angle to find the angles in these quadrants. The angles in the third and fourth quadrants are $\frac{\pi}{3} + \pi = \frac{4\pi}{3}$ and $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$ , respectively.

Q5: How do we express the answer in both radians and degrees?

A5: We have found the angles in the third and fourth quadrants that satisfy the equation $\csc \theta = -\frac{2 \sqrt{3}}{3}$ . To express the answer in both radians and degrees, we need to convert the angles from radians to degrees. Using the conversion factor $\pi \text{ radians} = 180^\circ$ , we get:

$\frac{4\pi}{3} \text{ radians} = \frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ$
$\frac{2\pi}{3} \text{ radians} = \frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ$

Q6: What are the final answers?

A6: The final answers are $\boxed{\frac{4\pi}{3}, \frac{2\pi}{3}}$ .