MEASURES of ANGLES: RADIANS FALL, 2015 DR. SHILDNECK

r s Radian Measure One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. 2 θ s = r θ = 1 radian

One revolution around a circle of radius r corresponds to 2 π radians because So, for an arclength of the entire circle (the circumference), Thus, there are 2 π radians around a circle. In other words, you can think of the central angle as "how many radii would it take to make up the arc length." A radian is defined as the angle equal to the ratio of the arclength to the radius: θ = s/r 3

4 Using this formula for radians allows us to solve any problem that requires us to find the central angle, radius or arclength. θ = s/r θ r = s r = s / θ We will use this tomorrow...

Converting Degrees to Radians and Radians to Degrees Since there are 2 π radians in a cirlce, and a circle consists of 360 o we can conclude that the two angle measurements are equivalent (but different units). Thus, in order to convert from Degree measure to Radian mesure (and vice­versa), you simply need to remember one simple rule: 360 degrees is the same as 2 π radians, and hence, 180 degrees is the same as π radians. 180 o = π radians,so, 180 o = 1 or π π = o Thus, to convert from degrees to radians: Take the degree measure and multiply by 1 in the form π 180 o To convert from radians to degrees: Take the radian measure and multiply by 1 in the form 180 o π 5

Convert from degrees to radians, or radians to degrees o 2. 6

Special Angles In the rotational system, there are two series of angles that are considered "special." Not coincidentally, these angles are related to our special right triangles. Thus, the patterns are those of 30 degrees and 45 degrees. You need to be able to find and mark these special angles, in their location, as you rotate around the coordinate axes. 7

Special Angles 0o0o 90 o 180 o 270 o 360 o 30 o 45 o 150 o 135 o 225 o 240 o 330 o 315 o 60 o 120 o 210 o 300 o

Special Angles 0 2π/4 4π/4 6π/4 8π/4 π/6 π/4 5π/6 3π/4 5π/4 8π/6 11π/6 7π/4 2π/64π/6 7π/6 10π/6 π/2 π 3π/2 2π2π (π/3) (2π/3) (4π/3) (5π/3) 3π/6 6π/6 9π/6 12π/6

Sketch each angle in standard position

11 A ssignment Worksheet 7 P. 238 #10‐17, 55‐56, 58‐61