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4.1 Radian and Degree measure Changing Degrees to Radians Linear speed Angular speed.

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Presentation on theme: "4.1 Radian and Degree measure Changing Degrees to Radians Linear speed Angular speed."— Presentation transcript:

1 4.1 Radian and Degree measure Changing Degrees to Radians Linear speed Angular speed

2 Definition of an angle An angle is made from two rays with a common initial point. In standard position the initial side is on the x axis

3 Positive angle vs. Negative angle Positive angles are Counter clockwise C.C.W. Negative angles are ClockwiseC.W.

4 Angles with the same initial side and terminal side are coterminal.

5

6 The measure of an angle is from initial side to terminal side Vertex at the origin (Center)

7 Definition of a Radian Radian is the measure of the arc of a unit circle. Unit circle is a circle with a radius of 1.

8 The quadrants in terms of Radians What is the circumference of a circle with radius 1?

9 The quadrants in terms of Radians What is the circumference of a circle with radius 1?

10 The quadrants in terms of Radians The circumference can be cut into parts.

11 The quadrants in terms of Radians The circumference can be cut into parts.

12 Find the Coterminal Angle Since equals 0. it can be added or subtracted from any angle to find a coterminal angle. Given

13 Complementary Angles – two angles are complementary if their sum is 90 degrees or Supplementary Angles have a sum of 180 degrees or Find the complementary and supplementary angles for

14 Radian vs. Degree measurements 360º = 180º = So or

15 Radian vs. Degree measurements 360º = 180º = So or To convert Degrees into Radians multiply by To convert Radians into Degrees multiply by

16 Conversions: Radians Degrees To convert degrees to radians, multiply by To convert radians to degrees, multiply by Converting an angle from to decimal form.

17 Change 140º to Radians Change to degrees Usedegree to rads. Userads to degrees

18 How to use radian to find Arc length The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle 360º In degrees Are length called S would be

19 How to use radian to find Arc length In degrees Are length called S would be In radian the equation is

20 r = 9, θ = 215ºChanging to rads Are length S

21 Linear speed and Angular speed Linear speed is Angular speed is Assuming constant speed

22 Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v Moreover, if is the angle (in radian measure) corresponding to the arc length s, then the angular speed(the lowercase Greek letter omega) of the particle is Angular speed A relationship between linear speed and angular speed is

23 Area of a Sector of a Circle where is measured in radians A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of 120 degrees. Find the area of the fairway watered by the sprinkler. 70 ft 120 o = how many radians? odd, 107

24 Finding Linear Speed The second hand of a clock is 10.2 cm long. Find the linear speed of the Tip of the second hand as it passes around the clock face. Linear speed v Arc length s =1.068 cm/sec


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