M1410402Part2

M1410402Part2 - 4.21 PART C: EXTENDING FROM QUADRANT I TO...

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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 non-quadrantal standard angle is the acute angle that its terminal side makes with the x-axis. 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 half-revolution 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 half-revolution 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 half-revolution 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 half-revolution 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.

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M1410402Part2 - 4.21 PART C: EXTENDING FROM QUADRANT I TO...

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