2. Find the lengths of both circular arcs of the unit circle connecting the point (–V2/2, -V2/2) and the endpoint of the radius the makes an angle of 8 radians with the positive hori- zontal axis.
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Problem 2: Calculating Arc Lengths on a Unit Circle**
Given: Determine the lengths of both circular arcs on the unit circle connecting two specific points:
1. The point is given as \((- \sqrt{2}/2, - \sqrt{2}/2)\).
2. The endpoint of a radius that makes an angle of 8 radians with the positive horizontal axis (x-axis).
Objective: Calculate the lengths of the minor and major circular arcs connecting the specified points on the unit circle.
**Steps to Solve:**
1. **Coordinate Identification:**
- Identify the coordinates \((- \sqrt{2}/2, - \sqrt{2}/2)\) on the unit circle. Given that the unit circle has a radius of 1, this point lies on the circle where both x and y coordinates are \(- \sqrt{2}/2\).
2. **Angle Calculation:**
- Compute the angle formed at the center of the circle between the point and the radius ending at 8 radians.
- Convert angle in radians within the range \([0, 2\pi]\) to identify the corresponding point on the circle.
3. **Arc Length Formula:**
- Use the arc length formula for the unit circle \( s = r\theta \), where \(r=1\) (radius of the unit circle) and \(\theta\) is the central angle.
- Calculate minor and major arc lengths connecting the two points using relevant angles.
**Example Diagram:**
- This problem involves an understanding of trigonometric functions and the geometry of the circle. Although no graph is provided here, drawing a unit circle with labeled axes helps visualize the calculations and angles involved.
By following these steps and correctly applying trigonometric principles, students can accurately determine the lengths of both circular arcs on the unit circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0fedb7-adbc-492c-aa14-9f5d12342848%2F9b55e66e-b66b-4617-ab11-272a26e3793e%2Ff2ek9hd_processed.png&w=3840&q=75)
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