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Question 1186939: Suppose that O=(0,0), A=(3.4,0), and B=(2.81,1.92). The arc between points B and C is 12.92 units long.
A. What is the value of (theta)1, the radian measure of Angle AOB?
B. What is the value of (theta)2, the radian measure of Angle BOC?
C. What are the x- and y-coordinates of point C?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's the solution, broken down step by step:
**A. θ₁ (Angle AOB):**
1. **Find the lengths of OA and OB:**
* OA = √((3.4 - 0)² + (0 - 0)²) = 3.4
* OB = √((2.81 - 0)² + (1.92 - 0)²) = √(7.8961 + 3.6864) = √11.5825 ≈ 3.403
2. **Use the dot product formula:**
* OA • OB = |OA| * |OB| * cos(θ₁)
* (3.4 * 2.81) + (0 * 1.92) = 3.4 * 3.403 * cos(θ₁)
* 9.554 = 11.5702 * cos(θ₁)
* cos(θ₁) = 9.554 / 11.5702 ≈ 0.8257
3. **Solve for θ₁:**
* θ₁ = arccos(0.8257) ≈ 0.5994 radians
**B. θ₂ (Angle BOC):**
1. **Find the radius of the circle:** Since OA and OB are radii, and we are working with a circular arc BC, OA is the radius. r = 3.4
2. **Use the arc length formula:**
* Arc length (s) = r * θ₂
* 12.92 = 3.4 * θ₂
* θ₂ = 12.92 / 3.4 ≈ 3.8 radians
**C. Coordinates of Point C:**
1. **Find the angle from the x-axis to OC:** This is θ₁ + θ₂ ≈ 0.5994 + 3.8 = 4.3994 radians.
2. **Use the coordinates formula:**
* x = r * cos(θ₁ + θ₂) = 3.4 * cos(4.3994) ≈ -1.0468
* y = r * sin(θ₁ + θ₂) = 3.4 * sin(4.3994) ≈ -3.2348
**Therefore:**
* θ₁ (Angle AOB) ≈ 0.5994 radians
* θ₂ (Angle BOC) ≈ 3.8 radians
* Coordinates of point C ≈ (-1.0468, -3.2348)
Answer by ikleyn(52858)  (Show Source):
You can put this solution on YOUR website! .
Strictly saying, this problem is posed INCORRECTLY,
since it does not define that point 'C' lies in the same circle as points 'A' and 'B'.
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