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Chapter 3 Radian Measure and Circular Functions

Chapter 3 Radian Measure and Circular Functions. Section 3.1 Radian Measure. We have seen that angles can be measured in degrees. In more theoretical work in mathematics, radian measure of angles is preferred.

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Chapter 3 Radian Measure and Circular Functions

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  1. Chapter 3Radian Measure and Circular Functions

  2. Section 3.1 Radian Measure We have seen that angles can be measured in degrees. In more theoretical work in mathematics, radian measure of angles is preferred. Radian measure does not use any artificial unit, it only uses the ratio of arc length to radius. arc length Radian measure of θ = θ r Since the circumference of a circle is 2πr, 360° = 2π radians

  3. 1 radian is Degrees Radians Radians Degrees Converting Between Degrees and Radians 360° = 2πradians 180° = π radians 90° = π/2 radians 1° = π/180 radians

  4. Section 3.2 Applications Radian Measure Example Find the length of the arc s when the radius and angle are given as in the diagram.

  5. Example Reno, Nevada is approx. due north of Los Angeles. The latitude of Reno is 40°N, while that of Los Angeles is 34°N. The radius of the Earth is 6400km. Find the north-south distance between the two cities.

  6. Example Find the area of that sector-shaped piece of land.

  7. Section 3.3 The Unit Circle and Circular Functions A unit circle is a circle whose radius is 1 unit. If we replace the angle θ (in radians) in the trigonometric functions by the arc length s on the unit circle, where s can be any real number, we get some new functions called circular functions.

  8. x-coordinate is cos(s) y-coordinate is sin(s)

  9. distance time angle θ Angular Speed is defined to be: ω = time Section 3.4 Linear and Angular Speed Linear Speed is defined to be: v =

  10. Example A satellite traveling in a circular orbit 1600km above the surface of Earth takes 2 hr to make an orbit. The radius of the Earth is approx. 6400km. Find the linear speed of the satellite. Example It takes Jupiter 11.64 yr to complete one orbit around the sun. If Jupiter’s average distance from the sun is 483,600,000 miles, find its linear speed in miles per second.

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