Select The Correct Answer.What Is The Corresponding Point On The Unit Circle For The Given Radian Measure $\theta=\frac{7 \pi}{6}$?A. $\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right$\] B. $\left(\frac{1}{2},

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 of sine, cosine, and tangent, and it plays a crucial role in solving problems involving right triangles and circular functions.

What is a Radian Measure?

A radian measure is a way of measuring angles in terms of the ratio of the arc length to the radius of a circle. It is defined as the ratio of the arc length to the radius of the circle. In other words, if an arc of a circle subtends an angle of $\theta$ radians, then the length of the arc is equal to $\theta$ times the radius of the circle.

Converting Degrees to Radians

To convert a degree measure to a radian measure, we can use the following formula:

$$\theta(\text{radians}) = \frac{\theta(\text{degrees})}{180} \times \pi$$

For example, if we want to convert 30 degrees to radians, we can use the formula above:

$$\theta(\text{radians}) = \frac{30}{180} \times \pi = \frac{\pi}{6}$$

The Given Radian Measure

In this problem, we are given a radian measure of $\theta = \frac{7\pi}{6}$. We need to find the corresponding point on the unit circle.

Finding the Corresponding Point on the Unit Circle

To find the corresponding point on the unit circle, we can use the following steps:

1. Determine the Quadrant: First, we need to determine the quadrant in which the point lies. Since the radian measure is $\frac{7\pi}{6}$, which is greater than $\frac{3\pi}{2}$, the point lies in the fourth quadrant.
2. Find the Reference Angle: Next, we need to find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis. In this case, the reference angle is $\frac{\pi}{6}$.
3. Find the Coordinates: Finally, we can find the coordinates of the point using the reference angle. Since the point lies in the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative.

Calculating the Coordinates

Using the reference angle of $\frac{\pi}{6}$, we can calculate the coordinates of the point as follows:

• x-coordinate: The x-coordinate is equal to the cosine of the reference angle, which is $\cos\left(\frac{\pi}{6}\right) = \frac{1}{2}$.
• y-coordinate: The y-coordinate is equal to the negative of the sine of the reference angle, which is $-\sin\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.

Therefore, the coordinates of the point on the unit circle corresponding to the radian measure $\theta = \frac{7\pi}{6}$ are $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.

Conclusion

In this problem, we used the unit circle and radian measures to find the corresponding point on the unit circle for the given radian measure $\theta = \frac{7\pi}{6}$. We determined the quadrant in which the point lies, found the reference angle, and calculated the coordinates of the point using the reference angle. The coordinates of the point on the unit circle corresponding to the radian measure $\theta = \frac{7\pi}{6}$ are $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$.

Answer

The correct answer is:

• A. $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$

The unit circle and radian measures are fundamental concepts in mathematics, particularly in trigonometry and geometry. In this article, we will answer some of the most frequently asked questions about the unit circle and radian measures.

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. It is used to define the trigonometric functions of sine, cosine, and tangent, and it plays a crucial role in solving problems involving right triangles and circular functions.

Q: What is a radian measure?

A: A radian measure is a way of measuring angles in terms of the ratio of the arc length to the radius of a circle. It is defined as the ratio of the arc length to the radius of the circle.

Q: How do I convert degrees to radians?

A: To convert a degree measure to a radian measure, you can use the following formula:

$$\theta(\text{radians}) = \frac{\theta(\text{degrees})}{180} \times \pi$$

Q: How do I find the corresponding point on the unit circle for a given radian measure?

A: To find the corresponding point on the unit circle for a given radian measure, you need to follow these steps:

1. Determine the Quadrant: First, you need to determine the quadrant in which the point lies. This can be done by looking at the radian measure and determining which quadrant it falls in.
2. Find the Reference Angle: Next, you need to find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis.
3. Find the Coordinates: Finally, you can find the coordinates of the point using the reference angle.

Q: What is the reference angle?

A: The reference angle is the acute angle between the terminal side of the angle and the x-axis. It is used to find the coordinates of the point on the unit circle.

Q: How do I find the coordinates of the point on the unit circle?

A: To find the coordinates of the point on the unit circle, you need to use the reference angle. The x-coordinate is equal to the cosine of the reference angle, and the y-coordinate is equal to the negative of the sine of the reference angle.

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

A: The unit circle is used to define the trigonometric functions of sine, cosine, and tangent. The values of these functions can be found by looking at the coordinates of the point on the unit circle.

Q: How do I use the unit circle to solve problems involving right triangles and circular functions?

A: To use the unit circle to solve problems involving right triangles and circular functions, you need to follow these steps:

1. Draw the Unit Circle: First, draw the unit circle and label the coordinates of the points on the circle.
2. Identify the Angle: Next, identify the angle that you are working with and determine the quadrant in which it lies.
3. Find the Reference Angle: Find the reference angle, which is the acute angle between the terminal side of the angle and the x-axis.
4. Find the Coordinates: Finally, find the coordinates of the point on the unit circle using the reference angle.

Q: What are some common applications of the unit circle and radian measures?

A: The unit circle and radian measures have many common applications in mathematics and science. Some of these applications include:

• Trigonometry: The unit circle is used to define the trigonometric functions of sine, cosine, and tangent.
• Geometry: The unit circle is used to solve problems involving right triangles and circular functions.
• Physics: The unit circle is used to describe the motion of objects in circular motion.
• Engineering: The unit circle is used to design and analyze circular systems, such as gears and pulleys.

Conclusion

In this article, we have answered some of the most frequently asked questions about the unit circle and radian measures. We have discussed the definition of the unit circle, the concept of radian measures, and how to use the unit circle to solve problems involving right triangles and circular functions. We have also discussed some common applications of the unit circle and radian measures.