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

The unit circle is a fundamental concept in mathematics, particularly in trigonometry and calculus. 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, in terms of the coordinates of a point on the circle.

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 at the center of the circle, 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:

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

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 Point on the Unit Circle

To find the point on the unit circle corresponding to the given radian measure, we can use the following steps:

1. Determine the Quadrant: The radian measure $\theta = \frac{7 \pi}{6}$ lies in the fourth quadrant of the unit circle.
2. Find the Reference Angle: The reference angle for $\theta = \frac{7 \pi}{6}$ is $\frac{\pi}{6}$.
3. Find the Coordinates: The coordinates of the point on the unit circle corresponding to the reference angle $\frac{\pi}{6}$ are $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Conclusion

Therefore, the corresponding point on the unit circle for the given radian measure $\theta = \frac{7 \pi}{6}$ is $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Answer

The correct answer is:

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

Discussion

This problem requires a good understanding of the unit circle and radian measures. The student needs to be able to convert degree measures to radian measures and find the corresponding point on the unit circle. The student also needs to be able to determine the quadrant and find the reference angle for a given radian measure.

Related Topics

• Unit Circle: The unit circle is a fundamental concept in mathematics, particularly in trigonometry and calculus. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
• Radian Measures: A radian measure is a way of measuring angles in terms of the ratio of the arc length to the radius of a circle.
• Trigonometric Functions: The trigonometric functions, such as sine, cosine, and tangent, are defined in terms of the coordinates of a point on the unit circle.

Practice Problems

• Find the corresponding point on the unit circle for the given radian measure $\theta = \frac{3 \pi}{4}$.
• Find the reference angle for the radian measure $\theta = \frac{5 \pi}{3}$.
• Determine the quadrant for the radian measure $\theta = \frac{11 \pi}{6}$.

Conclusion

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 a fundamental concept in mathematics, particularly in trigonometry and calculus.

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 a degree measure to a radian measure?

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

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

Q: What is the reference angle for a given radian measure?

A: The reference angle for a given radian measure is the acute angle between the terminal side of the angle and the x-axis.

Q: How do I find the reference angle for a given radian measure?

A: To find the reference angle for a given radian measure, you can use the following steps:

1. Determine the Quadrant: Determine the quadrant in which the radian measure lies.
2. Find the Absolute Value: Find the absolute value of the radian measure.
3. Find the Reference Angle: Find the reference angle by subtracting the absolute value from the radian measure.

Q: What is 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 can use the following steps:

1. Determine the Quadrant: Determine the quadrant in which the radian measure lies.
2. Find the Reference Angle: Find the reference angle for the radian measure.
3. Find the Coordinates: Find the coordinates of the point on the unit circle corresponding to the reference angle.

Q: How do I determine the quadrant for a given radian measure?

A: To determine the quadrant for a given radian measure, you can use the following steps:

1. Find the Absolute Value: Find the absolute value of the radian measure.
2. Determine the Quadrant: Determine the quadrant in which the absolute value lies.

Q: What are some common radian measures?

A: Some common radian measures include:

• $\frac{\pi}{6}$
• $\frac{\pi}{4}$
• $\frac{\pi}{3}$
• $\frac{\pi}{2}$
• $\frac{2 \pi}{3}$
• $\frac{3 \pi}{4}$
• $\frac{5 \pi}{6}$

Q: How do I use the unit circle to evaluate trigonometric functions?

A: To use the unit circle to evaluate trigonometric functions, you can use the following steps:

1. Determine the Quadrant: Determine the quadrant in which the radian measure lies.
2. Find the Reference Angle: Find the reference angle for the radian measure.
3. Find the Coordinates: Find the coordinates of the point on the unit circle corresponding to the reference angle.
4. Evaluate the Trigonometric Function: Evaluate the trigonometric function using the coordinates of the point on the unit circle.

Conclusion

In conclusion, the unit circle and radian measures are fundamental concepts in mathematics, particularly in trigonometry and calculus. The student needs to have a good understanding of these concepts to solve problems involving the unit circle and radian measures. The student also needs to be able to convert degree measures to radian measures and find the corresponding point on the unit circle.