solve this---**Question Number:** 19
**Chart Description:**
* **Type:** Geometric figure on a Cartesian coordinate system.
* **Main Elements:**
* **Coordinate Axes:** Horizontal x-axis pointing right, vertical y-axis pointing up. Both axes intersect at the origin O.
* **Origin:** Labeled O, located at the intersection of the axes, which is the center of the circle.
* **Circle:** A circle centered at O is shown.
* **Points:**
* O: Origin, center of the circle.
* A: A point on the circle with coordinates $(\sqrt{3}, 1)$.
* B: A point on the positive x-axis, seemingly on the circle.
* **Lines:**
* Line segment OA: Connects the origin O to point A.
* Line segment OB: Connects the origin O to point B (part of the positive x-axis).
* **Angles:** Angle $\angle AOB$ is formed by the line segments OA and OB.
**Textual Information:**
* **Question Stem:** In the xy-plane above, O is the center of the circle, and the measure of $\angle AOB$ is $\frac{\pi}{a}$ radians.
* **Question:** What is the value of $a$?
* **Options:** None are provided in the image.
* **Other Relevant Text:** The number 19 is shown at the top left.
视频信息
答案文本
视频字幕
In this problem, we have a circle centered at the origin O in the xy-plane. Point A is located on the circle with coordinates root 3 comma 1. Point B is on the positive x-axis, also on the circle. We're told that the measure of angle AOB is pi over a radians, and we need to find the value of a.
To find the value of a, we need to determine the angle theta between OA and OB. We can use trigonometry. Since we know the coordinates of point A are root 3 comma 1, we can find the tangent of the angle. The tangent of theta equals y over x, which is 1 over root 3. Simplifying this, we get root 3 over 3. Using the inverse tangent function, we find that theta equals pi over 6 radians. Since the problem states that the angle is pi over a radians, we can conclude that a equals 6.
Now that we've found the angle theta equals pi over 6 radians, we can determine the value of a. The problem states that the angle measure is pi over a radians. So we can write the equation: pi over a equals pi over 6. Since pi is non-zero, we can equate the denominators, giving us a equals 6. Therefore, the value of a is 6, which is our final answer.
Let's summarize what we've learned. We found the angle between OA and OB using trigonometry. For point A with coordinates root 3 comma 1, we calculated the tangent of theta to be 1 over root 3, which simplifies to root 3 over 3. This gives us theta equals pi over 6 radians. Since the problem states that the angle measure is pi over a radians, we set up the equation pi over a equals pi over 6. Solving for a, we get a equals 6, which is our final answer.