MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions Section 4 – Radians, Arc Length, and Angular Speed.

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MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions Section 4 – Radians, Arc Length, and Angular Speed

Angle Measurement – Degrees  Full rotation from the positive x-axis to the positive x-axis is 360°. Why?  Therefore, rotation from the positive x-axis to … the positive y-axis is 90°. the negative x-axis is 180°. the negative y-axis is 270°.

Problem: Find the length of an arc of a circle of radius r that is intercepted by a central angle of degree measure . a  r r Circumference of the circle is 2  r. The arc is  /360 th of the circle. Therefore, a is  /360 th of 2  r.

Angle Measurement – Radians  The measure of the angle  in radians is the length of the arc intercepted by the angle, divided by the radius.   = a / r radians x 2 + y 2 = r 2 a  r r Since any circle produces the same results, a unit circle is used (i.e. r = 1).

Angle Measurement: Radians vs. Degrees  With a right angle … 90° =  / 2 radians  1  =  / 180 radians  a° = (  / 180)a radians  b radians = (180 /  )b° Circumference of a Unit Circle: 2  22 90° 1 x 2 + y 2 = 1

Angle Measurement: Radians vs. Degrees  Degree and DMS measurements are designated using the symbols °, ‘, and “. Examples:  25.47°  25°28’12”  Radian measurements do not use any symbols. Examples:   283  / 2000 NOTE: All of the examples above are the same measurement (third one is approximate).

DegreesRadians 0°0°0 30°  / 6 45°  / 4 60°  / 3 90°  / 2 Common Angles: Degrees vs. Radians First Quadrant

DegreesRadians 90°  / 2 120° 2  / 3 135° 3  / 4 150° 5  / 6 180°  Common Angles: Degrees vs. Radians Second Quadrant

DegreesRadians 180°  210° 7  / 6 225°  / 4 240° 4  / 3 270° 3  / 2 Common Angles: Degrees vs. Radians Third Quadrant

DegreesRadians 270° 3  / 2 300° 5  / 3 315° 7  / 4 330° 11  / 6 360° 22 Common Angles: Degrees vs. Radians Fourth Quadrant

The Unit Circle - Summarized

The Unit Circle – Summarized Axis Points

The Unit Circle – Summarized k  /6 Points

The Unit Circle – Summarized k  /4 Points

The Unit Circle – Summarized k  /3 Points

Other Angles  Negative Angles Go clockwise (e.g. -  /6 is the same point as 11  /6) Ordered pairs are the same  Angles Greater than 2  Repeats the same points  General Rule: “Add or subtract multiples of 2  to get the radian measure between 0 and 2 .”

Problem: Find the length of an arc of a circle of radius r that is intercepted by a central angle of radian measure . a  r r Circumference of the circle is 2  r. The arc is  /(2  ) th of the circle. Therefore, a is  /(2  ) th of 2  r.