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Angles of Rotation and Radian Measure In the last section, we looked at angles that were acute. In this section, we will look at angles of rotation whose measure can be any real number. An angle of rotation is formed by two rays with a common endpoint (called the vertex). terminal side initial side y The other ray is called the terminal side. x One ray is called the initial side. vertex Angles of Rotation and Radian Measure y terminal side vertex initial side x The measure of the angle is determined by the amount and direction of rotation from the initial side to the terminal side. The angle measure is positive if the rotation is counterclockwise, and negative if the rotation is clockwise. A full revolution (counterclockwise) corresponds to 360º. Angles of Rotation and Radian Measure This is a positive (counterclockwise) angle y x y This is a negative (clockwise) angle x Angles of Rotation That would be a 90º Angle y x y x That would be a 180º Angle Angles of Rotation That would be a 270º Angle y x y x That would be a 360º Angle Angles of Rotation An Angle of 120º in standard position y x y An Angle of -120º in standard position x Example: Draw an angle with the given measure in standard position. Then determine in which quadrant the terminal side lies. A. 210º b. –45º c. 510º 150º 210º –45º Terminal side is in Quadrant III Terminal side is in Quadrant IV 510º Terminal side is in Quadrant II Use the fact that 510º = 360º + 150º. So the terminal side makes 1 complete revolution and continues another 150º. 150º 510º 510º and 150º are called coterminal (their terminal sides coincide). An angle coterminal with a given angle can be found by adding or subtracting multiples of 360º. So if you are asked to find coterminal angles you can simply add 360 to the angle or subtract 360 from the angle Find two angles that are coterminal with 130º (one positive and one negative 130º + 360º = 490º 130º - 360º = -290º Complimentary and Supplementary Angles 2 angles that are complimentary add up to equal 90 degrees 2 angles that are supplementary add up to equal 180 degrees Find the supplement to an angle of 24º 180 – 24 = 156 Find the compliment to an angle of 24º 90 – 24 = 66 You can also measure angles in radians. One radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r. r r one radian Since the circumference of a circle is 2πr, there are 2π radians in a full circle. Degree measure and radian measure are therefore related by the following: 360º = 2π radians Conversion Between Degrees and Radians • To rewrite a degree measure in radians, multiply by π radians 180º • To rewrite a radian measure in degrees, multiply by 180º π radians Examples: a. 240º Rewrite each in radians b. –90º 4 240º = 240º • π –90º = –90º • π 180º 180º 3 = –π = 4π 2 3 c. 135º 3 135º = 135º • π 180º 4 = 3π 4 240º = 4π radians 3 135º = 3π radians 4 –90º = –π radians 2 Examples: a. 5π 8 Rewrite each in degrees b. 16π 5 5π = 5π • 180º 8 8 π = 112.5º 16π = 16π • 180º 5 5 π = 576º Two positive angles are complementary if the sum of their measures is π/2 radians (which is 90º) Two positive angles are supplementary if the sum of their measures is π radians (which is 180º). Example: Find the complement of = π The complement is π – π 8 2 8 = 4π – π = 3π 8 8 8 Example: Find the supplement of = 3π The supplement is π – 3π 5 5 = 5π – 3π 5= 2π 5 5