Trigonometry—Law of Sines

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Trigonometry—Law of Sines
Page 24 Law of Sines: sin A sin B sin C (derived from the new area formula) a b c B Proof: a c C A b Problems: x 60° 7 6 x 45° 8 60° y 9 55° 40° x y

Page 25 Radians—The Other Measurement for Angles radius arc central angle 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. What is the measure of angle  in radians? Ans: ________ 6 What is the relationship between , r and s? s How many radians do you think angle  is? r 6 12 8 12 4 2 Find the indicated variable: 2 rad. 4 s 1.8 rad. r 9 8 20 Ans: ______ _______ _______ Ans: _____ ______ ______

Page 26 What is the relationship between radians and degrees? Q1: How many degrees are there in a circle? A: 360 degrees 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, r 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 27  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 ____

Trigonometry—Area of a Sector and a Segment
Page 28 Area of a Sector 60° Area = ? O 4 Area of a Sector—Formulas /4 6 If  is in degrees: If  is in radians: A r Area of a Segment Area of a Segment—Formulas Area = ? Area = ? If  is in degrees: A 60° /4 O 4 O 6 O r If  is in radians:

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

cos (30 + 60) = cos 30 cos 60 – sin 30 sin 60
Trigonometry—Addition and Subtraction Identities Page 30 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 (30 + 60) = cos 30 cos 60 – sin 30 sin 60 ? 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. Note: 1. The real reason I am not doing the proof is because it’s long and tedious (and worst of all, you probably won’t get it anyway).

cos ( + ) = cos  cos  – sin  sin 
Trigonometry—Proving Addition and Subtraction Identities Page 31 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: Proof:

a) sin ( + ) b) cos ( – ) c) tan ( + )
Trigonometry—Applying Addition and Subtraction Identities Page 32 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):  sin  = 3/5  cos  =  tan  = For :  cos  = –5/13  sin  =  tan  =