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

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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 author’s 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 π 6 radians. Brothers include coterminal “twins.” For example, 30 (or π 6 radians) is a “twin” for the 330 (or 11 π 6 radian) angle.
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4.22 Let’s try to discover patterns from the four “famous” positive brothers of the π 6 angle (including π 6 , itself) noted in the figure on the previous page . Quadrant II: π π 6 = 6 π 6 1 π 6 = 5 ! 6 We get this angle by making a half-revolution counterclockwise (corresponding to π radians) and then backtracking (going clockwise) by π 6 radians. Trick: 5 is 1 less than 6. Quadrant III: π + π 6 = 6 π 6 + 1 π 6 = 7 ! 6 We get this angle by making a half-revolution counterclockwise (corresponding to π radians) and then proceeding counterclockwise by another π 6 radians. Trick: 7 is 1 more than 6. Quadrant IV: 2 π π 6 = 12 π 6 1 π 6 = 11 ! 6 We get this angle by making a full revolution counterclockwise (corresponding to 2 π radians) and then backtracking (going clockwise) by π 6 radians. Trick: 11 is 1 less than twice 6. In fact, these patterns apply to any reference (acute) angle of the form π k radians, where k is an integer greater than 2.
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4.23 Here are the “famous” positive brothers of the π 6 , π 4 , and π 3 angles: We’ve already seen the situation with π 6 : (The boxes correspond to Quadrants.) 5 π 6 π 6 7 π 6 11 π 6 Now, π 4 : 3 π 4 π 4 5 π 4 7 π 4 Now, π 3 : 2 π 3 π 3 4 π 3 5 π 3 Coterminal “twins” have the same trig values, including the signs. This is true, because coterminal standard angles correspond to the same intersection point P cos θ , sin θ ( ) on the unit circle. For example, π 6 and 11 π 6 have the same sin, cos, tan, csc, sec, and cot values. Their common intersection point is 3 2 , 1 2 .
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4.24 Why is it useful to deal with brothers? Brothers have the same basic trig values up to (i.e., except maybe for) the signs. In other words, the basic trig values between two brothers are the same in magnitude, or absolute value.
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