Presentation on theme: "Trigonometry—Law of Sines"— Presentation transcript:
1Trigonometry—Law of Sines Page 24Law of Sines: sin A sin B sin C (derived from the new area formula)a b cBProof:acCAbProblems:x60°76x45°860°y955°40°xy
2Trigonometry—Radian Measure Page 25Radians—The Other Measurement for Anglesradiusarccentral angleHow 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: ________6What is the relationship between , r and s?sHow many radians do you think angle is?r61281242Find the indicated variable:2 rad.4s1.8 rad.r9820Ans: ______ _______ _______Ans: _____ ______ ______
3Trigonometry—Radians and Degree Conversions Page 26What is the relationship between radians and degrees?Q1: How many degrees are there in a circle?A: 360 degreesrQ2: 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,rConclusion: 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.
4 rad. = 180° Trigonometry—Radians and Degree Conversions (cont’d) 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 ____
5Trigonometry—Area of a Sector and a Segment Page 28Area of a Sector60°Area = ?O4Area of a Sector—Formulas/46If is in degrees:If is in radians:ArArea of a SegmentArea of a Segment—FormulasArea = ?Area = ?If is in degrees:A60°/4O4O6OrIf is in radians:
6sin = Conclusion: cos = sin (–) = tan = cos (–) = tan (–) = Trigonometry—Negative-Angle IdentitiesPage 29sin =cos =tan =Conclusion:sin (–) =cos (–) =tan (–) =csc (–) =sec (–) =cot (–) =ryx––ysin (–) =cos (–) =tan (–) =r
7cos (30 + 60) = cos 30 cos 60 – sin 30 sin 60 Trigonometry—Addition and Subtraction IdentitiesPage 30Addition Identities Subtraction Identitiescos ( + ) = 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 thatcos ( + ) = 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).
8cos ( + ) = cos cos – sin sin Trigonometry—Proving Addition and Subtraction IdentitiesPage 31Addition Identities Subtraction Identitiescos ( + ) = 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 thatcos ( + ) = 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:
9a) sin ( + ) b) cos ( – ) c) tan ( + ) Trigonometry—Applying Addition and Subtraction IdentitiesPage 32If sin = 3/5 (/2 < < ) and cos = –5/13 ( < < 3/2), finda) 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 =