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1、3.3 两角和与差及二倍角公式一【复习要求】1.掌握两角和与差的正弦、余弦、正切公式，了解它们的内在联.2.掌握二倍角的正弦、余弦、正切公式.2.能够利用两角和与差的公式、二倍角公式进行三角函数式的求值、化简和证明.二、【知识回顾】1两角和与差的三角函数 ； ； ； ； ； ；2二倍角公式：在中令，可得相应的二倍角公式。 ； = = 。3降幂公式 ； .注意：二倍角公式具有“升幂缩角“作用，降幂公式具有“降幂扩角”作用4辅助角公式，（其中不能同时为0）证明： 其中，且角终边过点 在使用时，不必死记结论，而重在这种收缩（合二为一）思想如： ； 。5.公式的使用技巧（1）连续应用：（2）“1”的代
2、换：， （3）收缩代换：，（其中不能同时为0）（4）公式的变形：如： 。 。（5）角的变换（拆角与配角技巧）， ， ， ， ， ，（6）二倍角公式的逆用及常见变形 二倍角的正用、逆用、变形应用是公式的三种主要使用方法，特别是二倍角的余弦公式，它在求值、化简、证明中有广泛的应用，解题时应根据不同的需要，灵活选取。；5三角函数式的化简（1）化简方法：直接应用公式进行降次、消项；化切为弦，异名化同名，异角化同角； 三角公式的逆用等。降幂或升幂（2）化简要求：能求出值的应求出值；使三角函数种数尽量少；使项数尽量少；尽量使分母不含三角函数；尽量使被开方数不含三角函数。6三角函数的求值类型有三类（1）给角
3、求值：一般所给出的角都是非特殊角，要观察所给角与特殊角间的关系，利用三角变换消去非特殊角，转化为求特殊角的三角函数值问题；（2）给值求值：给出某些角的三角函数式的值，求另外一些角的三角函数值，解题的关键在于“变角”，如等，把所求角用含已知角的式子表示，求解时要注意角的范围的讨论；（3）给值求角：实质上转化为“给值求值”问题，关键也在于“变角”，把所求角用含已知角的式子表示，由所得的函数值结合所求角的范围或函数的单调性求得角。7三角等式的证明（1）三角恒等式的证明根据等式两端的特征，通过三角恒等变换，应用化繁为简、左右同一、转换命题等方法，使等式两端化“异”为“同”；（2）三角条件等式的证明通过
4、观察，发现已知条件和待证等式间的关系。若从结论开始，通过变形，将已知表达式代入得出结论，采用代入法；若从条件开始，化简条件，将其代入要证表达式中，通过约分抵消等消去某些项，从而得出结论，采用消参法；若这两种方法都证不出来，可采用分析法进行证明。三【例题精讲】考点一、给角求值例.求值：(2)sin(75)cos(45)cos(15)【反思归纳】对于给角求值问题，往往所给角都是非特殊角，解决这类问题的基本思路有： 化为特殊角的三角函数值 化为正负相消的项，消去求值 化分子、分母使之出现公约数进行约分而求值。考点二、给值求值例3.已知，求的值.例4.已知，求的值考点三、给值求角探究给值求值问题(已知
5、某角的三角函数值，求另一角的三角函数值)例2已知0，cos，sin，求sin()的值变式迁移2已知tan2，tan .(1)求tan 的值；(2)求的值来源:学,科,网例3已知0，tan ，cos().(1)求sin 的值；(2)求的值变式迁移3若sin A，sin B，且A、B均为钝角，求AB的值考点四、三角函数式的化简与证明例.已知，且（1） 化简（2） 是否存在，使与相等？若存在，求出；若不存在，说明理由。例(14分)已知向量a(cos ，sin )，b(cos ，sin )，|ab|.(1)求cos()的值；(2)若0，且sin ，求sin 的值【练习】1. 已知，则 2. 求值： 3
6、. 在中，已知，则的值为 4. （08年高考山东卷改编）已知，则= 5. （07年高考江苏卷）若，则 6. （08年江苏卷）如图，在平面直角坐标第中，以轴为始边作两个锐角，它们的终边分别与单位圆相交于A、B两点，已知A、B的横坐标分别为，（1）求的值；（2）求的值待添加的隐藏文字内容37. 已知为锐角，向量，.(1) 若,求角的值;(2) 若,求的值.8. 若,且都是锐角,求9. (2010淮安调研,16)已知,.(1) 若,求的值.(2) 若,求的值.Winger Tuivasa-Sheck, who scored two tries in the Kiwis 20-18 semi-fina
7、l win over England, has been passed fit after a lower-leg injury, while Slater has been named at full-back but is still recovering from a knee injury aggravated against USA.Both sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charnleys effort in t
8、heir first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competitions top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named Rugby League International Federation player of the year Sonny Bill Williams who has attracted the m
9、ost interest in the tournament so far.The Kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the World Cup in both rugby league and rugby union after triumphing with the All Blacks in 2011.Id give every award back in a heartbeat just to get across the line
10、 this weekend, said Williams.The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifas anti-racism taskforce, is in London for the Football Associations 150th anniversary celebrations and will attend Citys Premier League match at Chelsea on Sunday.I am going to be at the match t
11、omorrow and I have asked to meet Yaya Toure, he told BBC Sport.For me its about how he felt and I would like to speak to him first to find out what his experience was.Uefa hasopened disciplinary proceedings against CSKAfor the racist behaviour of their fans duringCitys 2-1 win.Michel Platini, presid
12、ent of European footballs governing body, has also ordered an immediate investigation into the referees actions.CSKA said they were surprised and disappointed by Toures complaint. In a statement the Russian side added: We found no racist insults from fans of CSKA. Baumgartner the disappointing news:
13、 Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached,
14、will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the s
15、tratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.They blew it in 2008 when the
16、y got caught cold in the final and they will not make the same mistake against the Kiwis in Manchester.Five years ago they cruised through to the final and so far history has repeated itself here - the last try they conceded was scored by Englands Josh Charnley in the opening game of the tournament.
17、That could be classed as a weakness, a team under-cooked - but I have been impressed by the Kangaroos focus in their games since then.They have been concentrating on the sort of stuff that wins you tough, even contests - strong defence, especially on their own goal-line, completing sets and a good k
18、ick-chase. Theyve been great at all the unglamorous stuff that often goes unnoticed in the stands but not by your team-mates.It is as though their entire tournament has been preparation for the final.In Johnathan Thurston, Cooper Cronk, Cameron Smith and either Billy Slater or Greg Inglis at full-ba
19、ck they have a spine that is unmatched in rugby league. They have played in so many high-pressure games - a priceless asset going into Saturday.The Kiwis are a lot less experienced but winning a dramatic match like their semi-final against England will do wonders for their confidence.They defeated Australia in the Four Nations final in 2010 and the last World Cup, and know they can rise to the big occasion.
