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Unformatted text preview: 4.21 PART C: EXTENDING FROM QUADRANT I TO OTHER QUADRANTS Reference angles The reference angle for a nonquadrantal standard angle is the acute angle that its terminal side makes with the xaxis. Brothers (the authors term) are angles that have the same reference angle. For example, the angles below are brothers; they all have the same reference angle, namely 30 , or 6 radians. Brothers include coterminal twins. For example, 30 (or 6 radians) is a twin for the 330 (or 11 6 radian) angle. 4.22 Lets try to discover patterns from the four famous positive brothers of the 6 angle (including 6 , itself) noted in the figure on the previous page . Quadrant II: 6 = 6 6 1 6 = 5 ! 6 We get this angle by making a halfrevolution counterclockwise (corresponding to radians) and then backtracking (going clockwise) by 6 radians. Trick: 5 is 1 less than 6. Quadrant III: + 6 = 6 6 + 1 6 = 7 ! 6 We get this angle by making a halfrevolution counterclockwise (corresponding to radians) and then proceeding counterclockwise by another 6 radians. Trick: 7 is 1 more than 6. Quadrant IV: 2 6 = 12 6 1 6 = 11 ! 6 We get this angle by making a full revolution counterclockwise (corresponding to 2 radians) and then backtracking (going clockwise) by 6 radians. Trick: 11 is 1 less than twice 6. In fact, these patterns apply to any reference (acute) angle of the form k radians, where k 4.22 Lets try to discover patterns from the four famous positive brothers of the 6 angle (including 6 , itself) noted in the figure on the previous page . Quadrant II: 6 = 6 6 1 6 = 5 ! 6 We get this angle by making a halfrevolution counterclockwise (corresponding to radians) and then backtracking (going clockwise) by 6 radians. Trick: 5 is 1 less than 6. Quadrant III: + 6 = 6 6 + 1 6 = 7 ! 6 We get this angle by making a halfrevolution counterclockwise (corresponding to radians) and then proceeding counterclockwise by another 6 radians. Trick: 7 is 1 more than 6. Quadrant IV: 2 6 = 12 6 1 6 = 11 ! 6 We get this angle by making a full revolution counterclockwise (corresponding to 2 radians) and then backtracking (going clockwise) by 6 radians. Trick: 11 is 1 less than twice 6. In fact, these patterns apply to any reference (acute) angle of the form k radians, where k is an integer greater than 2. 4.23 Here are the famous positive brothers of the 6 , 4 , and 3 angles: Weve already seen the situation with 6 : (The boxes correspond to Quadrants.) 5 6 6 7 6 11 6 Now, 4 : 3 4 4 5 4 7 4 Now, 3 :...
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.
 Fall '11
 staff
 Calculus, Angles

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