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8.1. RADIANS AND ARC LENGTH. 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. Arc length = 1. radius =1. 1 radian. Relationship Between Radians and Degrees.

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8.1

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  1. 8.1 RADIANS AND ARC LENGTH Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  2. 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. Arc length = 1 radius =1 1 radian Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  3. 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) ≈ 57.296◦. • Thus, one radian is approximately 57.296◦. • One-quarter revolution, or 90◦, is equal to ¼(2 π) or π/2 radians. • Since π ≈ 3.142, one complete revolution is about 6.283 radians and one-quarter revolution is about 1.571 radians. Equivalences for Common Angles Measured in Degrees and Radians Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  4. 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≈ 57.296◦ and 1◦ = π/180 ≈ 0.01745 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

  5. Converting Between Degrees and Radians Example 3 • Convert 3 radians to degrees. • Convert 3 degrees to radians. Solution (a) 3 radians・180◦/(π radians) = 540◦/(π radians) ≈ 171.887◦. (b) 3◦・π radians/180◦= π radians/60 ≈ 0.052 radians. The word “radians” is often dropped, so if an angle or rotation is referred to without units, it is understood to be in radians. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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

  7. 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 ● s = 4 miles s = 4 miles Θ=4 rad Θ=4/3 rad ● ● 1 mile 3 miles ● Arc length 4 and radius 1, so angle θ = s/1= 4 radians Arc length 4 and radius 3, so angle θ = s/r= 4/3 radians Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  8. 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◦ = 0.9985. 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 = −0.99999873. This is reasonable, because 3.14 radians is extremely close to π radians or 180◦, so cos 3.14 ≈ cosπ = −1. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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