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Section 6.1 Angle Measure

Chapter 6 – Trigonometric Functions: Right Triangle Approach. Section 6.1 Angle Measure. Definitions. Line: Line Segment: Ray: . Angles. An angle is formed by two rays with a common vertex. One of the rays is called the initial side and the other ray is called the terminal side.

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Section 6.1 Angle Measure

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  1. Chapter 6 – Trigonometric Functions: Right Triangle Approach Section 6.1 Angle Measure 6.1 - Angle Measure

  2. Definitions • Line: • Line Segment: • Ray: 6.1 - Angle Measure

  3. Angles An angle is formed by two rays with a common vertex. One of the rays is called the initial side and the other ray is called the terminal side. Terminal Side Vertex Initial Side 6.1 - Angle Measure

  4. Definitions • Measure The measure of an angle is the amount of rotation about the vertex required to move from the initial side to the terminal side. • Degree One unit of measurement for angles is the degree. An angle of 1 degree is formed by rotating the terminal side 1/360 of a complete revolution. 6.1 - Angle Measure

  5. Definitions • In calculus and other branches of mathematics, a more natural method of measuring angles is used – radian measure. • Radian Measure If a circle of radius 1 is drawn with the vertex of an angle at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends (is opposite to) the angle. 6.1 - Angle Measure

  6. Radian Measure 6.1 - Angle Measure

  7. Radians and Degrees 6.1 - Angle Measure

  8. Radians and Degrees • To understand the size of a radian, notice that 6.1 - Angle Measure

  9. Examples – pg. 440 • Find the radian measure of the angle with the given degree measure. 4. 54o 7. -75o • Find the degree measure of the angle with the given radian measure. 6.1 - Angle Measure

  10. Angle in Standard Position • An angle in standard position is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis. 6.1 - Angle Measure

  11. Coterminal Angles • Two angles in standard position are coterminal if their sides coincide. 6.1 - Angle Measure

  12. Finding Coterminal Angles Coterminal angles can be found by adding or subtracting 360° from . For radians, coterminal angles can be found by adding or subtracting 2 from . 6.1 - Angle Measure

  13. Examples – pg. 440 • Find an angle between 0o and 360o that is coterminal with the given angle. 43. -800o 44. 1270o • Find an angle between 0 and 2that is coterminal with the given angle. 6.1 - Angle Measure

  14. Arc Length • In a circle of radius r, the length s of an arc that subtends a central angle of  radians is s = r Note: We can solve this formula for  and get 6.1 - Angle Measure

  15. Examples – pg. 440 56. A central angle  in a circle of radius 5 m is subtended by an arc of length 6 m. Find the measure of  in degrees and in radians. 58. A circular arc of length 3 ft subtends a central angle of 25o.Find the radius of the circle. 6.1 - Angle Measure

  16. Area of a Circular Sector • In a circle of radius r, the area Aof a sector with a central angle of  radians is 6.1 - Angle Measure

  17. Examples – pg. 440 65. The area of a sector of a circle with a central angle of 2 rad is 16 m2. Find the radius of the circle. 6.1 - Angle Measure

  18. Linear Speed • Linear speed is the rate at which the distance traveled is changing. • That means, linear speed is the distance traveled divided by the time elapsed. 6.1 - Angle Measure

  19. Solving for velocity we get . Our distance when in a circle is the arc length so we have . Linear Speed We know the distance formula is d = rt. Rate is the same as velocity so d = vt. . 6.1 - Angle Measure

  20. Angular Velocity (Speed) • Angular speed is the rate at which the central angle is changing, so angular speed is the number of radians this angle changes divided by the time elapsed. 6.1 - Angle Measure

  21. Angular Velocity (Speed) The angular velocityis the angle, , (that is the Greek letter “omega”) generated in one unit of time by a line segment from the center of the circle to a point, P, on the circumference. The angle and is defined by Note:  must be measured in radians 6.1 - Angle Measure

  22. Linear and Angular Speed - Summary 6.1 - Angle Measure

  23. Relationship between Linear and Angular Speed • If a point moves along a circle radius r with angular speed, , then its linear speed is given by v = r 6.1 - Angle Measure

  24. How? We can start with our arc length formula s=r and divide both sides by t. The angular speed is in radians/time 6.1 - Angle Measure

  25. Note • If angular velocity is given as revs/time, then we must convert it to radians/time before we start our problem. 6.1 - Angle Measure

  26. Examples – pg. 442 6.1 - Angle Measure

  27. Examples – pg. 442 6.1 - Angle Measure

  28. Examples – pg. 442 6.1 - Angle Measure

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