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TrigonometryLaw of Sines Law of Sines: sin A sin B sin C (derived from the new area formula) abc Proof: b a c B AC Problems: x 60° 76 x 45° 8 60° y 9 55° 40° x y Page 24

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TrigonometryRadian Measure RadiansThe Other Measurement for Angles How big is 1 radian? A (central) angle sustains a measure of 1 radian if the length of the intercepted arc is exactly the same as the length of the radius. 6 6 radius arc central angle What is the measure of angle in radians? Ans: ________ 6 12 How many radians do you think angle is? Ans: ____________________ r s What is the relationship between, r and s? 2 rad. 4 s Find the indicated variable: Ans: _____ ______ _ _____ 1.8 rad. r Page 25

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TrigonometryRadians and Degree Conversions What is the relationship between radians and degrees? Q1: How many degrees are there in a circle? A: 360 degrees r r Q2:How many radians are there in a circle? A:We know that = s/r. If we go around the whole circle, s, the arc, becomes the circumference, which is denoted by __. So, Conclusion: In degrees, there are ____ in a circle; in radians, there are ____ radians in a circle. Therefore, ___ = ___ rad. Divide both sides by 2, we obtain ___ = ___ rad. Page 26

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TrigonometryRadians and Degree Conversions (contd) rad. = 180° 1 rad. = ( )° 1 rad. ____° ( ) rad. = 1° 1° ____ rad. Common angles in radians and degrees: = 180° = 90° = 60° = 45° = 30° In general, a) How to convert radians to degrees: Multiply by ____ b) How to convert degrees to radians: Multiply by ____ Page 27

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TrigonometryArea of a Sector and a Segment Area of a Sector 60° Area = ? O4 Area of a SectorFormulas /4 Area = ? O6 If is in degrees: If is in radians: A Or Area of a Segment Area = ? O4 Area of a SegmentFormulas Area = ? O6 If is in degrees: If is in radians: Or 60° /4 A Page 28

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TrigonometryNegative-Angle Identities – sin (– ) = cos (– ) = tan (– ) = sin = cos = tan = x y r Conclusion: sin (– ) = cos (– ) = tan (– ) = csc (– ) = sec (– ) = cot (– ) = –y r Page 29

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TrigonometryAddition and Subtraction Identities Addition Identities Subtraction Identities cos ( + ) = cos cos – sin sin cos ( – ) = cos cos + sin sin sin ( + ) = sin cos + cos sin sin ( – ) = sin cos – cos sin tan ( + ) = tan ( – ) = Michael Sullivan, the author of the textbook, used a full page (page 409) to prove that cos ( + ) = cos cos – sin sin, which I am not going to do the proof here. (1) What I am going to do is to verify the identity (or formula) is true if I use = 30 and = 60. That is, cos ( ) = cos 30 cos 60 – sin 30 sin 60 Note: 1. The real reason I am not doing the proof is because its long and tedious (and worst of all, you probably wont get it anyway). ? The real application: Find the value of the following without using a calculator: 1. cos 37 cos 53 – sin 37 sin 53 = 2. sin 94 cos 49 – cos 94 sin 49 = 3. sin 88 cos 62 + cos 88 sin 62 = 4. Page 30

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TrigonometryProving Addition and Subtraction Identities Addition Identities Subtraction Identities cos ( + ) = cos cos – sin sin cos ( – ) = cos cos + sin sin sin ( + ) = sin cos + cos sin sin ( – ) = sin cos – cos sin tan ( + ) = tan ( – ) = I am (still) not going to prove that cos ( + ) = cos cos – sin sin but I am going prove some of the other ones here, based on the fact that we are going to take the above identity for granted (i.e., accepting it to be true without knowing the proof). We will also need some of our already-proven identities, namely, the cofunction identities and the negative-angle identities: From the Subtraction Identities:From the Addition Identities: cos ( – ) = cos cos + sin sin sin ( + ) = sin cos + cos sin Proof: Page 31

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TrigonometryApplying Addition and Subtraction Identities If sin = 3/5 ( / 2 < < ) and cos = –5/13 ( < < 3 / 2 ), find a) sin ( + ) b) cos ( – ) c) tan ( + ) For : Solution: a) sin ( + ) = b) cos ( – ) = c) tan ( + ) = Alternate Solution (approximate): a) sin ( + ) = b) cos ( – ) = c) tan ( + ) = sin = 3/5 cos = tan = cos = –5/13 sin = tan = Page 32

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