Glossary

Angles and circular motion are key concepts in math, linking geometry and . They help us understand how objects move in circles and how to measure rotations, which is useful in many real-world applications.

We'll look at different ways to measure angles, convert between units, and calculate arc lengths. We'll also explore circular motion, including linear and , and how these concepts relate to the and trigonometry.

Angles and Circular Motion

Angles in standard position

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  • of positioned at origin with extending along positive x-axis
  • determined by angle's measure
    • Positive angles measured counterclockwise from initial side (90°, 180°)
    • Negative angles measured clockwise from initial side (-45°, -120°)
  • Quadrantal angles have terminal side along coordinate axes
    • 0° or 0 radians: points along positive x-axis
    • 90° or π2\frac{\pi}{2} radians: points along positive y-axis
    • 180° or π\pi radians: points along negative x-axis
    • 270° or 3π2\frac{3\pi}{2} radians: points along negative y-axis

Degree and radian conversions

  • Radians provide alternative unit for measuring angles
    • One equals angle subtended by equal to radius of circle
  • Converting degrees to radians: multiply angle in degrees by π180\frac{\pi}{180}
    • 60° = 60 × π180\frac{\pi}{180} = π3\frac{\pi}{3} radians
    • 135° = 135 × π180\frac{\pi}{180} = 3π4\frac{3\pi}{4} radians
  • Converting radians to degrees: multiply angle in radians by 180π\frac{180}{\pi}
    • π6\frac{\pi}{6} radians = π6\frac{\pi}{6} × 180π\frac{180}{\pi} = 30°
    • 5π4\frac{5\pi}{4} radians = 5π4\frac{5\pi}{4} × 180π\frac{180}{\pi} = 225°

Coterminal angles

  • Angles sharing the same terminal side
  • Finding coterminal angles in degrees: add or subtract multiples of 360° to given angle
    • 45° and 405° are coterminal (45° + 360° = 405°)
    • -30° and 330° are coterminal (-30° + 360° = 330°)
  • Finding coterminal angles in radians: add or subtract multiples of 2π\pi to given angle
    • π3\frac{\pi}{3} and 7π3\frac{7\pi}{3} are coterminal (π3\frac{\pi}{3} + 2π\pi = 7π3\frac{7\pi}{3})
    • −π4-\frac{\pi}{4} and 7π4\frac{7\pi}{4} are coterminal (−π4-\frac{\pi}{4} + 2π\pi = 7π4\frac{7\pi}{4})

Arc length calculation

  • Arc length proportional to measure of
  • Formula: s=rθs = r\theta
    • ss: arc length
    • rr: radius of circle
    • θ\theta: central angle in radians (convert from degrees if necessary)
  • Circle with radius 8 units and central angle of 90°
    • Convert angle to radians: 90° × π180\frac{\pi}{180} = π2\frac{\pi}{2} radians
    • Calculate arc length: s=8×π2=4π≈12.57s = 8 \times \frac{\pi}{2} = 4\pi \approx 12.57 units
  • Circle with radius 3 units and central angle of 2π3\frac{2\pi}{3} radians
    • Calculate arc length: s=3×2π3=2π≈6.28s = 3 \times \frac{2\pi}{3} = 2\pi \approx 6.28 units

Circular motion concepts

  • (vv): speed of object moving along circular path
    • Formula: v=2πrTv = \frac{2\pi r}{T}
      • rr: radius of circle
      • TT: (time for one complete )
  • Angular velocity (ω\omega): rate of change of angular position
    • Formulas: ω=2πT\omega = \frac{2\pi}{T} or ω=vr\omega = \frac{v}{r}
    • Measured in radians per unit time
  • Relationship between linear and angular velocity: v=rωv = r\omega
  • Ferris wheel with radius 20 m and period of 30 seconds
    • Calculate linear velocity: v=2π×2030≈4.19v = \frac{2\pi \times 20}{30} \approx 4.19 m/s
    • Calculate angular velocity: ω=2π30≈0.21\omega = \frac{2\pi}{30} \approx 0.21 rad/s
  • Car wheel with radius 0.3 m and angular velocity of 10 rad/s
    • Calculate linear velocity: v=0.3×10=3v = 0.3 \times 10 = 3 m/s

Trigonometry and the Unit Circle

  • Trigonometry: branch of mathematics dealing with relationships between sides and angles of triangles
  • Unit circle: circle with radius 1 centered at the origin, used to define trigonometric functions
  • : angular movement around a fixed point, measured in degrees or radians
  • Revolution: complete 360° rotation around a fixed point
  • : distance around a circle, calculated using the formula C = 2πr, where r is the radius
  • : mathematical constant representing the ratio of a circle's circumference to its diameter, approximately 3.14159

Key Terms to Review (29)

Addition method: The addition method, also known as the elimination method, is a technique used to solve systems of linear equations. It involves adding or subtracting the equations to eliminate one variable, simplifying the system to a single equation with one variable.
Angle: An angle is the figure formed by two rays, called the sides of the angle, that share a common endpoint called the vertex. Angles are fundamental geometric concepts that are essential in understanding various mathematical topics, including 7.1 Angles and 10.1 Non-right Triangles: Law of Sines.
Angle of rotation: The angle of rotation is the angle through which a figure or point is rotated about a fixed point, typically the origin. It is measured in degrees or radians.
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object. It describes the speed at which an object rotates or revolves around a fixed axis or point, and is a fundamental concept in the study of rotational motion.
Arc Length: Arc length is the distance measured along a curved path, such as a segment of a circle or an ellipse. It is a fundamental concept in geometry and calculus, with applications in various fields, including engineering, physics, and computer graphics.
Central Angle: A central angle is an angle formed by two radii of a circle that intersect at the center of the circle. It represents the measure of the arc that the angle subtends on the circle's circumference.
Circumference: Circumference is the distance around the outside of a circular or cylindrical object. It represents the perimeter of a circle and is a fundamental concept in geometry and trigonometry.
Co-vertex: The co-vertices of an ellipse are the endpoints of the minor axis. They are perpendicular to and lie at the midpoint of the major axis.
Cost function: A cost function represents the cost of producing a certain number of goods or services as a function of the quantity produced. It is typically expressed in algebraic form and used to model economic behavior.
Coterminal Angle: A coterminal angle is an angle that has the same terminal side as another angle, but a different initial side. In other words, coterminal angles are angles that share the same position on the coordinate plane, differing only by a multiple of 360 degrees or 2\u03C0 radians.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
Factor by grouping: Factor by grouping is a method used to factor polynomials that involves rearranging and combining terms into groups that have a common factor. This technique is particularly useful for polynomials with four or more terms.
Independent system: An independent system is a set of linear equations with exactly one solution. The graphs of these equations intersect at a single point.
Initial Side: The initial side of an angle is one of the two rays that form the angle. It is the reference line from which the angle is measured, typically oriented in the positive horizontal direction.
Linear Velocity: Linear velocity is the rate of change in the position of an object along a straight line. It describes the speed of an object's motion in a specific direction and is a vector quantity, meaning it has both magnitude and direction.
Period: The period of a function is the distance or interval along the independent variable axis over which the function's shape or pattern repeats itself. It is a fundamental concept in the study of periodic functions, such as trigonometric functions, and is essential for understanding their properties and graphs.
Pi (π): Pi (π) is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific applications, including the study of real numbers, angles, trigonometry, and the graphing of trigonometric functions.
Profit function: A profit function, often denoted as $P(x)$, represents the difference between total revenue and total costs for a given number of units sold or produced, $x$. It is a key concept in economic analysis and optimization.
Quadrantal Angle: A quadrantal angle is an angle that measures a specific fraction of a full circle, typically one of the four 90-degree angles formed by the coordinate axes. These angles are important in the study of trigonometry and the unit circle.
Radian: A radian is a unit of angle measurement in mathematics, representing the angle subtended by an arc on a circle that is equal in length to the radius of that circle. It is a fundamental unit in trigonometry, providing a way to measure angles that is independent of the size of the circle.
Revenue function: A revenue function calculates the total revenue generated from selling a certain number of units of a product. It is usually expressed as $R(x) = p \cdot x$, where $p$ is the price per unit and $x$ is the number of units sold.
Revolution: A revolution is a complete or radical change in a situation or in the way things are done. In the context of angles, a revolution refers to the complete rotation of an angle around a fixed point, typically measured in degrees or radians.
Rotation: Rotation is the circular motion of an object around a fixed axis or point. It is a fundamental concept in mathematics and physics that describes the movement of an object as it turns around a central point or line.
System of linear equations: A system of linear equations consists of two or more linear equations involving the same set of variables. The solutions to the system are the points where the graphs of these equations intersect.
Terminal Side: The terminal side of an angle is the ray that forms the end or endpoint of the angle. It is one of the two rays that define the angle, with the other being the initial side.
Trigonometry: Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It provides the tools and techniques to analyze and solve problems involving angles, lengths, and the relationships between them.
Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) of the coordinate plane. It is a fundamental concept in trigonometry, as it provides a visual representation of the relationships between the trigonometric functions and the angles they represent.
Vertex: The vertex is a critical point in various mathematical functions and geometric shapes. It represents the point of maximum or minimum value, or the point where a curve changes direction. This term is particularly important in the context of quadratic equations, functions, absolute value functions, and conic sections such as the ellipse and parabola.
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