Find The Terminal Point On The Unit Circle Determined By $\frac{4\pi}{3}$ Radians. Use Exact Values, Not Decimal Approximations.

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Introduction

The unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The unit circle is used to define the trigonometric functions, such as sine, cosine, and tangent, which are essential in various mathematical and scientific applications. In this article, we will explore how to find the terminal point on the unit circle determined by a given angle in radians.

Understanding the Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The equation of the unit circle is $x^2 + y^2 = 1$, where $x$ and $y$ are the coordinates of a point on the circle. The unit circle is divided into four quadrants, each with a specific range of angles.

Angles in Radians

Angles can be measured in degrees or radians. Radians are a more fundamental unit of measurement, as they are based on the ratio of the arc length to the radius of a circle. The radian measure of an angle is defined as the ratio of the arc length to the radius of the circle. In this article, we will use radians to measure angles.

Finding the Terminal Point

To find the terminal point on the unit circle determined by a given angle in radians, we need to use the trigonometric functions, such as sine and cosine. The sine and cosine functions are defined as the ratios of the opposite side to the hypotenuse and the adjacent side to the hypotenuse, respectively.

Using the Sine and Cosine Functions

The sine and cosine functions are defined as follows:

• $\sin(\theta) = \frac{y}{r}$
• $\cos(\theta) = \frac{x}{r}$

where $\theta$ is the angle in radians, $x$ and $y$ are the coordinates of the terminal point, and $r$ is the radius of the circle.

Finding the Terminal Point for $\frac{4\pi}{3}$ Radians

To find the terminal point on the unit circle determined by $\frac{4\pi}{3}$ radians, we need to use the sine and cosine functions. We can start by finding the value of $\sin(\frac{4\pi}{3})$ and $\cos(\frac{4\pi}{3})$.

Calculating the Sine and Cosine Values

Using the unit circle and the definition of the sine and cosine functions, we can calculate the values of $\sin(\frac{4\pi}{3})$ and $\cos(\frac{4\pi}{3})$.

• $\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}$
• $\cos(\frac{4\pi}{3}) = -\frac{1}{2}$

Finding the Terminal Point

Now that we have the values of $\sin(\frac{4\pi}{3})$ and $\cos(\frac{4\pi}{3})$, we can find the terminal point on the unit circle determined by $\frac{4\pi}{3}$ radians.

• $x = r \cos(\theta) = 1 \cdot (-\frac{1}{2}) = -\frac{1}{2}$
• $y = r \sin(\theta) = 1 \cdot (-\frac{\sqrt{3}}{2}) = -\frac{\sqrt{3}}{2}$

Conclusion

In this article, we have explored how to find the terminal point on the unit circle determined by a given angle in radians. We have used the sine and cosine functions to calculate the values of $\sin(\frac{4\pi}{3})$ and $\cos(\frac{4\pi}{3})$, and then used these values to find the terminal point on the unit circle determined by $\frac{4\pi}{3}$ radians. The terminal point is located at $(-\frac{1}{2}, -\frac{\sqrt{3}}{2})$.

Applications of the Unit Circle

The unit circle has many applications in mathematics and science. It is used to define the trigonometric functions, which are essential in various mathematical and scientific applications. The unit circle is also used in the study of periodic functions, such as the sine and cosine functions, which are used to model periodic phenomena in physics and engineering.

Final Thoughts

In conclusion, the unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is used to define the trigonometric functions, which are essential in various mathematical and scientific applications. The unit circle is also used in the study of periodic functions, such as the sine and cosine functions, which are used to model periodic phenomena in physics and engineering. By understanding the unit circle and how to find the terminal point on the unit circle determined by a given angle in radians, we can gain a deeper understanding of the mathematical and scientific principles that govern our world.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by I.M. Yaglom

Further Reading

• [1] "The Unit Circle" by Khan Academy
• [2] "Trigonometry" by MIT OpenCourseWare
• [3] "Calculus" by Stanford University

Note: The references and further reading sections are not included in the word count.

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Introduction

In our previous article, we explored how to find the terminal point on the unit circle determined by a given angle in radians. We used the sine and cosine functions to calculate the values of $\sin(\frac{4\pi}{3})$ and $\cos(\frac{4\pi}{3})$, and then used these values to find the terminal point on the unit circle determined by $\frac{4\pi}{3}$ radians. In this article, we will answer some frequently asked questions (FAQs) related to finding the terminal point on the unit circle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The equation of the unit circle is $x^2 + y^2 = 1$, where $x$ and $y$ are the coordinates of a point on the circle.

Q: What are the sine and cosine functions?

A: The sine and cosine functions are defined as the ratios of the opposite side to the hypotenuse and the adjacent side to the hypotenuse, respectively. They are used to calculate the values of $x$ and $y$ for a given angle in radians.

Q: How do I find the terminal point on the unit circle determined by a given angle in radians?

A: To find the terminal point on the unit circle determined by a given angle in radians, you need to use the sine and cosine functions. You can start by finding the value of $\sin(\theta)$ and $\cos(\theta)$, and then use these values to calculate the coordinates of the terminal point.

Q: What is the relationship between the unit circle and the trigonometric functions?

A: The unit circle is used to define the trigonometric functions, such as sine and cosine. The values of $\sin(\theta)$ and $\cos(\theta)$ are used to calculate the coordinates of the terminal point on the unit circle.

Q: Can I use the unit circle to find the terminal point on a circle with a radius other than 1?

A: Yes, you can use the unit circle to find the terminal point on a circle with a radius other than 1. You just need to multiply the coordinates of the terminal point by the radius of the circle.

Q: What are some common angles in radians that I should know?

A: Some common angles in radians that you should know include:

• $\frac{\pi}{6}$ (30 degrees)
• $\frac{\pi}{4}$ (45 degrees)
• $\frac{\pi}{3}$ (60 degrees)
• $\frac{\pi}{2}$ (90 degrees)
• $\frac{2\pi}{3}$ (120 degrees)
• $\frac{3\pi}{4}$ (135 degrees)
• $\frac{5\pi}{6}$ (150 degrees)
• $\frac{7\pi}{6}$ (210 degrees)
• $\frac{4\pi}{3}$ (240 degrees)

Q: How do I use the unit circle to find the terminal point on the unit circle determined by a given angle in radians?

A: To use the unit circle to find the terminal point on the unit circle determined by a given angle in radians, you need to follow these steps:

1. Draw the unit circle and label the origin as (0, 0).
2. Draw a line from the origin to the point on the unit circle that corresponds to the given angle in radians.
3. Label the coordinates of the terminal point as (x, y).
4. Use the sine and cosine functions to calculate the values of $\sin(\theta)$ and $\cos(\theta)$.
5. Use the values of $\sin(\theta)$ and $\cos(\theta)$ to calculate the coordinates of the terminal point.

Q: What are some real-world applications of the unit circle?

A: The unit circle has many real-world applications, including:

• Calculating the height of a building or a mountain
• Determining the distance between two points on a map
• Calculating the area of a circle or an ellipse
• Modeling periodic phenomena in physics and engineering

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to finding the terminal point on the unit circle. We have also provided some common angles in radians that you should know and some real-world applications of the unit circle. By understanding the unit circle and how to find the terminal point on the unit circle determined by a given angle in radians, you can gain a deeper understanding of the mathematical and scientific principles that govern our world.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by I.M. Yaglom

Further Reading

• [1] "The Unit Circle" by Khan Academy
• [2] "Trigonometry" by MIT OpenCourseWare
• [3] "Calculus" by Stanford University