388 CHAPTER 6 Trigonometric Functions From this, we have 1 revolution 2 radians π = (5) 180 radians π ° = (6) Since 1revolution 360 , = ° we have 360 2 radians π ° = Dividing both sides by 2 yields Divide both sides of equation (6) by 180. Then 1degree 180 radian π = Divide both sides of equation (6) by .π Then 180 degrees 1 radian π = We have the following two conversion formulas:* 1degree 180 radian 1 radian 180 degrees π π = = (7) Example 4, parts (a)–(d), illustrates that angles that are “nice” fractions of a revolution are expressed in radian measure as fractional multiples of ,π rather than as decimals. For example, a right angle, as in Example 4(d), is left in the form 2 π radians, which is exact, rather than using the approximation 2 3.1416 2 1.5708 radians. π ≈ = When the fractions are not “nice,” use the decimal approximation of the angle, as in Example 4(e). Converting from Degrees to Radians Convert each angle in degrees to radians. (a) 60° (b) 150° (c) 45 − ° (d) 90° (e) 107° Solution (a) π π ° = ⋅ = ⋅ = 60 60 1 degree 60 180 radian 3 radians (b) π π ° = ⋅ ° = ⋅ = 150 150 1 150 180 radian 5 6 radians (c) π π − ° = − ⋅ = − 45 45 180 radian 4 radian (d) π π ° = ⋅ = 90 90 180 radian 2 radians (e) π ° = ⋅ ≈ 107 107 180 radian 1.868 radians Now Work PROBLEMS 23 AND 49 EXAMPLE 4 *Some students prefer instead to use the proportion π ° = Degrees 180 Radians . Then substitute for what is given and solve for the measurement sought. NOTE Because π is approximately 3.1416, the fact that π ° = 180 radians, confirms our suggestion from Figure 14(c). j
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