8.1 RADIANS AND ARC LENGTH Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Definition of a Radian An angle of 1 radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length 1. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally 1 radian Arc length = 1 radius =1

Relationship Between Radians and Degrees The circumference, C, of a circle of radius r is given by C = 2πr. In a unit circle, r = 1, so C = 2π. This means that the arc length spanned by a complete revolution of 360 ◦ is 2π, so 360 ◦ = 2 π radians. Dividing by 2π gives 1 radian = 360 ◦ /(2π) ≈ ◦. Thus, one radian is approximately ◦. One-quarter revolution, or 90 ◦, is equal to ¼(2 π) or π/2 radians. Since π ≈ 3.142, one complete revolution is about radians and one-quarter revolution is about radians. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Angle in degrees 0°30°45°60°90°120°135°150°180°270°360° 720° Angle in radians 0π/6π/4π/3π/22π/33π/45π/6π3π/22π2π 4π4π Equivalences for Common Angles Measured in Degrees and Radians

Converting Between Degrees and Radians To convert degrees to radians, or vice versa, we use the fact that 2π radians = 360 ◦. So 1 radian = 180 ◦ / π ≈ ◦ and 1 ◦ = π/180 ≈ radians. Thus, to convert from radians to degrees, multiply the radian measure by 180 ◦ /π radians. To convert from degrees to radians, multiply the degree measure by π radians/180 ◦. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Converting Between Degrees and Radians Example 3 (a)Convert 3 radians to degrees. (b)Convert 3 degrees to radians. Solution (a) 3 radians ・ 180 ◦ /(π radians) = 540 ◦ / (π radians) ≈ ◦. (b) 3 ◦ ・ π radians/180 ◦ = π radians/60 ≈ radians. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally The word “radians” is often dropped, so if an angle or rotation is referred to without units, it is understood to be in radians.

Arc Length in Circle of Radius r The arc length, s, spanned in a circle of radius r by an angle of θ in radians is s = r θ. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Converting Between Degrees and Radians Example 6 You walk 4 miles around a circular lake. Give an angle in radians which represents your final position relative to your starting point if the radius of the lake is: (a) 1 mile (b) 3 miles Solution Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Θ=4 rad Θ=4/3 rad Arc length 4 and radius 1, so angle θ = s/1= 4 radians (229°) Arc length 4 and radius 3, so angle θ = s/r= 4/3 radians (76°) ● ●● ● s = 4 miles 3 miles s = 4 miles 1 mile s = r θ

Sine and Cosine of a Number Example 7 Evaluate: (a) cos 3.14 ◦ (b) cos 3.14 Solution (a)Using a calculator in degree mode, we have cos 3.14 ◦ = This is reasonable, because a 3.14 ◦ angle is quite close to a 0 ◦ angle, so cos 3.14 ◦ ≈ cos 0 ◦ = 1. (b) Here, 3.14 is not an angle measured in degrees; instead we interpret it as an angle of 3.14 radians. Using a calculator in radian mode, we have cos 3.14 = − This is reasonable, because 3.14 radians is extremely close to π radians or 180 ◦, so cos 3.14 ≈ cos π = −1. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally