Radian Measure MHF4U: Advanced Functions In the past, we have - PDF document

t r i g o n o m e t r y t r i g o n o m e t r y Radian Measure MHF4U: Advanced Functions In the past, we have worked exclusively with degrees as our unit of measurement for angles. An alternative measurement system uses radians , rather than degrees. One radian is defined as the measure of the angle that is Radian Measure subtended by an arc that has a length equal to that of the radius. J. Garvin J. Garvin — Radian Measure Slide 1/14 Slide 2/14 t r i g o n o m e t r y t r i g o n o m e t r y Radian Measure Radian Measure We know that there are 360 ◦ in a circle, but how many Let r be the radius of a circle, and let a be the length of an arc that subtends an angle θ . radians is this? For one full rotation, the arc becomes the circumference. Since C = 2 π r , this implies that θ = 2 π r = 2 π . r So, there are 2 π (approximately 6 . 28) radians in a circle. Thus, there are π radians in a semi-circle, π 2 radians in a quarter-circle, etc. A relationship for these three quantities is θ = a r , where θ is measured in radians. J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 3/14 Slide 4/14 t r i g o n o m e t r y t r i g o n o m e t r y Radian Measure Converting Between Radians and Degrees How many degrees make up one radian, and vice versa? Set up a proportion as follows: d ◦ 360 ◦ 1 rad = 2 π rad d = 360 2 π = 180 π Therefore, one radian is 180 degrees. π π Using a similar method, one degree is 180 radians. We can use these values to convert between radians and degrees as necessary. J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 5/14 Slide 6/14

t r i g o n o m e t r y t r i g o n o m e t r y Converting Between Radians and Degrees Converting Between Radians and Degrees Example Example Convert 50 ◦ to radians. Convert 2 rad to degrees. 180 = 5 π π 2 × 180 50 × 18 ≈ 0 . 873 rad. π ≈ 114 . 6 ◦ . Example Example Convert 270 ◦ to radians. Convert 7 π 4 rad to degrees. 180 = 3 π π 270 × 2 ≈ 4 . 712 rad. 7 π 4 × 180 π = 1260 = 315 ◦ . 4 When possible, leave all angles in exact form to preserve accuracy. J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 7/14 Slide 8/14 t r i g o n o m e t r y t r i g o n o m e t r y Arc Length Arc Length One of the advantages of using radian measure rather than Example degrees is that it makes calculating arc length on a circle A bob is at the end of a pendulum with an arm length of 40 easier. cm. If the bob swings through an angle of 2 π 3 , determine the distance travelled by the bob through the air. Recall that one radian is defined as the measure of an angle that is subtended by an arc with a length equal to the circle’s radius. Rearranging the equation θ = a r for a , we obtain a = r θ . a = r θ This means that if we know a circle’s radius and the angle = 40 × 2 π that is formed by the subtended arc, we can calculate the 3 = 80 π length of the arc. 3 ≈ 83 . 8 cm J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 9/14 Slide 10/14 t r i g o n o m e t r y t r i g o n o m e t r y Arc Length Angular Velocity Example As an object rotates, its angular displacement changes with respect to time. An arc on the circumference of a circle with a diameter of 12 cm has a length of 22 . 8 cm. Determine the measure of the This rate of change is known as an object’s angular velocity . angle subtended by the arc. Angular Velocity Given an object’s angle of rotation, θ , and time, t , the Since the diameter is 12 cm, the radius is 6 cm. object’s angular velocity, ω , is given by ω = θ t . θ = 22 . 8 6 While it is not strictly necessary to express angular velocity in = 3 . 8 rad terms of radians, it typically is. J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 11/14 Slide 12/14

t r i g o n o m e t r y t r i g o n o m e t r y Angular Velocity Questions? Example A child rides a carousel that completes 20 revolutions in 2 minutes. Determine the child’s angular velocity. ω = 20 × 2 π 2 ω = 20 π rad/min J. Garvin — Radian Measure J. Garvin — Radian Measure Slide 13/14 Slide 14/14