What Is The Exact Value Of $\sin \left(-240^{\circ}\right$\]?A. $\frac{2 \pi}{3}$B. $-\frac{1}{2}$C. $\frac{\sqrt{3}}{2}$D. $-\frac{\sqrt{3}}{2}$

What is the Exact Value of $\sin \left(-240^{\circ}\right)$?

Understanding the Problem

The problem requires finding the exact value of the sine function for an angle of $-240^{\circ}$. This involves using trigonometric identities and properties to simplify the expression and arrive at the final answer.

Recalling Trigonometric Identities

To solve this problem, we need to recall some basic trigonometric identities, particularly the unit circle and the periodic properties of the sine function.

• The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
• The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• The sine function has a period of $360^{\circ}$, meaning that the value of the sine function repeats every $360^{\circ}$.

Converting the Angle to a Standard Position

To evaluate the sine function for an angle of $-240^{\circ}$, we need to convert it to a standard position, which is an angle between $0^{\circ}$ and $360^{\circ}$.

• Since the angle is negative, we can add $360^{\circ}$ to it to get a positive angle: $-240^{\circ} + 360^{\circ} = 120^{\circ}$.
• However, we need to consider the quadrant in which the angle lies. Since the angle is in the fourth quadrant, we need to take the negative of the sine value.

Using the Unit Circle

To find the exact value of the sine function for an angle of $120^{\circ}$, we can use the unit circle.

• The unit circle has a radius of 1, and the angle of $120^{\circ}$ is in the second quadrant.
• The sine function is positive in the second quadrant, so the value of the sine function for an angle of $120^{\circ}$ is $\frac{\sqrt{3}}{2}$.

Considering the Quadrant

Since the original angle is in the fourth quadrant, we need to take the negative of the sine value.

• Therefore, the exact value of the sine function for an angle of $-240^{\circ}$ is $-\frac{\sqrt{3}}{2}$.

Conclusion

In conclusion, the exact value of the sine function for an angle of $-240^{\circ}$ is $-\frac{\sqrt{3}}{2}$. This involves using trigonometric identities and properties to simplify the expression and arrive at the final answer.

Key Takeaways

• The sine function has a period of $360^{\circ}$.
• The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
• The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.

Final Answer

The final answer is $-\frac{\sqrt{3}}{2}$.
Q&A: Understanding the Exact Value of $\sin \left(-240^{\circ}\right)$

Frequently Asked Questions

Here are some frequently asked questions related to the exact value of the sine function for an angle of $-240^{\circ}$.

Q: What is the period of the sine function?

A: The period of the sine function is $360^{\circ}$, meaning that the value of the sine function repeats every $360^{\circ}$.

Q: How do I convert a negative angle to a standard position?

A: To convert a negative angle to a standard position, you can add $360^{\circ}$ to it. For example, to convert $-240^{\circ}$ to a standard position, you can add $360^{\circ}$ to get $120^{\circ}$.

Q: What is the value of the sine function for an angle of $120^{\circ}$?

A: The value of the sine function for an angle of $120^{\circ}$ is $\frac{\sqrt{3}}{2}$.

Q: Why do I need to take the negative of the sine value for an angle in the fourth quadrant?

A: Since the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants, you need to take the negative of the sine value for an angle in the fourth quadrant.

Q: What is the exact value of the sine function for an angle of $-240^{\circ}$?

A: The exact value of the sine function for an angle of $-240^{\circ}$ is $-\frac{\sqrt{3}}{2}$.

Q: Can I use the unit circle to find the exact value of the sine function for an angle of $-240^{\circ}$?

A: Yes, you can use the unit circle to find the exact value of the sine function for an angle of $-240^{\circ}$. The unit circle has a radius of 1, and the angle of $120^{\circ}$ is in the second quadrant. The sine function is positive in the second quadrant, so the value of the sine function for an angle of $120^{\circ}$ is $\frac{\sqrt{3}}{2}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $-\frac{\sqrt{3}}{2}$.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

• The sine function has a period of $360^{\circ}$.
• The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
• The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.

Q: How can I apply this knowledge to other problems?

A: You can apply this knowledge to other problems by using the unit circle and trigonometric identities to simplify expressions and arrive at the final answer.

Q: What are some common mistakes to avoid when solving trigonometry problems?

A: Some common mistakes to avoid when solving trigonometry problems are:

• Not converting negative angles to standard positions.
• Not considering the quadrant in which the angle lies.
• Not using the unit circle and trigonometric identities to simplify expressions.

Q: How can I practice and improve my skills in trigonometry?

A: You can practice and improve your skills in trigonometry by:

• Solving problems and exercises in trigonometry.
• Using online resources and tutorials to learn and practice trigonometry.
• Joining study groups or seeking help from teachers or tutors.