|
Besides degrees, another way to measure arcs is radians. This is an animation of 1 radian, 2, 3, π and 2π radians. What do you notice? Circles: Degrees and Radians What is a "radian"? An angle of 1 radian cuts off an arc equal to a length of 1 radius. An angle of 2π radians represents a full circle. 2π radians = degrees 1π radian = degrees 180 ? 360 ? If 2π radii are needed for a circle, 1π radii would wrap around half ? C=2πr It takes 2π radii to travel a full circle. full ? a circle. ★ Note! In upper level math,0º, and 0 radians isalways located "due east". We travel around the circle in a counterclockwise direction to move toward 360º or 2π radians Drag the radian measureto the correctlocation onthe circle. π rad ? π rad2 ? 3π rad 2 0 rad 2π rad You can use a conversion factor to convert any angle in degrees to the equivalent radian measure. An example is shown to convert 100º to radians. Simplify the fraction. Your answer should be in terms of π. 100º 1 × 100º = π rad180º 59 π = radians 100180 π radians Next, simplify your answer. Use a calculator to simplify the fraction. Remember to include π in your answer. Convert 120º to radians. First, drag the appropriate drop values. 120º 1 × π rad180º ? 120 180 = 120º = 120π 180 ? simplified ↑ π rad Simplify your answer. Simplify the fractionand remember to includeπ in your answer. Convert 70º to radians. First drag the appropriate drop values. 70º 1 × π rad180º ? = 70º = 70 π180 ? simplified ↑ π rad You can use a conversion factor to convert any angle in radians to the equivalent degree measure. An example is shown to convert 3π radians to degrees. The π's will cancel out of the numerator and denominator. Simplify the fraction. 3π rad 5 3π rad 5 × 180º π rad = 108º = 5 540º 5 Simplify your answer. Divide the numerator by the denominator. 5π rad 6 Convert 5π radians to degrees. × 6 180ºπ rad ? = 5π rad 6 900 6 ? º Multiply the numerators.Multiply the denominators.Don't include π in yourcalculations. The π's cancel out! = º Convert 3π radians to degrees. Simplify your answer. 3π rad 2 2 × 180ºπ rad ? = 3π rad 2 540 2 ? º = º |