解三角形讲义

1、解三角形【知识归类】1正弦定理：在abc中，注意：（1）r表示abc外接圆的半径；（2）正弦定理可以变形成各种形式来使用；（3）应用正弦定理解决的题型：已知两角和一边，求其它已知两边和一边的对角，求其它（4）在已知两边和一边的对角，求其它的类型中，可能出现无解、一解或两解，具体后面例题再进一步分析.2.余弦定理：在abc中，变形为：，注意：（1）应用余弦定理解决的题型：已知三边，求各角已知两边和一边的对角，求其它已知两边和夹角，求其它；（2）正、余弦定理的实质是一样的，从而正弦定理能解的问题余弦定理也一定能解，反之亦然；只是方便程度有别；（3）正、余弦定理可以结合使用；3.abc的面积公式，.

2、4.三角形形状的判定方法常用两种途径：（1）由正余弦定理将边转化为角；（2）由正余弦定理将角转化为边；注：化简中将三角形内角和、三角同角基本关系式、诱导公式、两角和与差的三角公式等综合结合起来.5.解三角形的应用可大体上把它分成以下三类：（1）距离问题：一点可到达另一点不可到达；两点都不可到达；（2）高度问题（最后都转化为解直角三角形）；（3）角度问题；（4）面积问题.【题型归类】题型一：正弦定理的应用例1 （1）在abc中， a=，b=，a=6,求c；（2）在abc中，a=，b=4，a=3,则cosb；（3）在abc中，b=，b=,c=150,求a.【审题要津】这里已知两角和一边或两边和一对

3、角，适合正弦定理解决的类型.解：（1）由正弦定理得（2）由正弦定理得又由三角函数同角基本关系得(3) 由正弦定理得或当时，当时，故或【方法总结】当已知两边和一对角，求其它时，可能有无解、一解或两解，注意讨论.题型二：余弦定理的应用例2（1）在abc中， a=2，b=，c=4,求b；（2）在abc中， a=7，b=3，c=5, 求最大角和sinc；（3）在abc中， a=，b=，,求c.【审题要津】这里已知两边和夹角或三边求其它，适合余弦定理解决的类型；当已知两边和一对角，求其它时也可使用余弦定理.解：（1）由余弦定理得，（2）为最大角.由余弦定理得又由正弦定理得（3）由余弦定理得，解得或【方法

4、总结】正、余弦定理的实质是一样的，从而正弦定理能解的问题余弦定理也一定能解，注意难易方法的选择.题型三：判定下列三角形的形状例3（1）在abc中，已知，判断abc的形状； （2）在abc中，已知，判断abc的形状； （3）在abc中，若，试判断abc的形状.【审题要津】这里已知边和角判断abc的形状，正确选择正、余弦定理进行解决.解：（1）由正弦定理得，由余弦定理得，故abc为钝角三角形.（2）,又故abc为等边三角形.（3）解法1由正弦定理得展开得故abc为等边三角形.解法2 由余弦定理得整理得故abc为等边三角形.【方法总结】常用两种途径：（1）由正余弦定理将边转化为角；（2）由正余弦定理

5、将角转化为边；注：化简中将三角形内角和、三角同角基本关系式、诱导公式、两角和与差的三角公式等综合结合起来.变式练习：设为钝角三角形的三边，求实数的取值范围？解：为钝角三角形的三边，解得此时最大，要使是三角形的三边，还需得设最长边所对的角为，则，解得故的取值范围为题型四：三角形的面积例4 （1）在abc中，已知，求abc的面积；（2）在abc中，面积求(3) 在abc中，边的长是方程的两个根，求边c的长.【审题要津】这里已知边和角求abc的面积或变形应用面积公式求解.解：（1）.(2) (3) 边的长是方程的两个根，【方法总结】根据三角形的面积知关键在于两邻边的乘积与夹角的正弦值的积，结合条件直

6、接应用或整体求解.题型五：三角形恒等式例5 在中，求证.【审题要津】由左边看出含有角和边，可由余弦定理将角的余弦值转化为边的比值，再由正弦定理化变为角的正弦；左边中分子、分母的边是齐次式可由正弦定理将边转化为角求的.证明：解法1 化角为边得：左边右边.故原等式成立.解法2化边为角得：左边右边.故原等式成立.【方法总结】解决此类题时，既要用到三角形有关的恒等式，又要用到任意角的三角函数的恒等式；证明时注意分析等式两边的形式是边还是角，便于从正余弦定理转化证得.题型六：应用题例6 如图，港口a北偏东方向的c处有一观测站，港口正东方向的b处有一轮船，测得bc为，该轮船从b处沿正西方向航行后到d处，测

7、得cd为，问此时轮船离港口a还有多远？【审题要津】要求ad的长，在中，只要求出即可，可由正弦定理求解；要求，可在中，由余弦定理求解.解：易知，设在中，由余弦定理得：在中，由正弦定理得：故此时轮船离港口a还有【方法总结】正余弦定理在实际应用中很广泛，常见题有：距离、高度、角度等问题；解决时，首先要认真分析题意，找出各量间的关系，根据题意画出示意图，将问题抽象为三角形模型（数学建模），然后利用正余弦定理求解，最后将结果转化为实际问题.【思想方法】1.数学思想：函数与方程、转化与化归、分类讨论的数学思想. 2.数学方法: 数学计算方法、公式法、转化与化归的方法、建模法等.习题一选择题：1在abc中，

8、若，则边（ d ）.（a）（b）（c）（d）.2在abc中，已知a=30，a=8，b=则三角形的面积为（d ）（a）（b）16（c）或16（d）或3在abc中，a、b、c所对边分别为a,b,c且，则a等于（ b ）（a）（b）（c）（d）4.在abc中，三边长ab=7，bc=5，ac=6，则的值为（ d ）（a）19（b）-14（c）-18（d）-195.若，则abc的形状为（ b ）.（a）等边三角形（b）等腰直角三角形（c）有一个角为30的直角三角形（d）有一个角为30的等腰三角形.6. 在直角三角形中，a、b为两锐角，则中（ b ）（a）有最大值和最小值0（b）有最大值和无最小值（c）无

9、最大值也无最小值（d）有最大值1，但无最小值.二、真空题：7.在abc中，若b=30，ab=，ac=2，则abc的面积为 或 .8. 已知三角形的三边成公差为2的等差数列，且它的最大角的正弦值为，则这个三角形的面积为 .9.若以2，3，为三边组成一个锐角三角形，则的取值范围是 .10.abc中，若ab=1，bc=2，则角c的取值范围是 .三、解答题： 11.已知是中角的对边，且，求这个三角形的最大内角.12.在abc中，bc=a，ac=b，a,b是方程的两个根，且.求（1）角c的度数；（2）ab的长；（3）abc的面积. （1）c=120（2）ab=（3）winger tuivasa-shec

10、k, who scored two tries in the kiwis 20-18 semi-final 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

11、 not conceded a try since josh charnleys effort in their 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

12、the year sonny bill williams who has attracted the most 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 a

13、ward back in a heartbeat just to get across the line 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

14、at chelsea on sunday.i am going to be at the match tomorrow 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 t

15、heir fans duringcitys 2-1 win.michel platini, president 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 fr

16、om fans of cska. baumgartner the disappointing news: 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 limit

17、ed cloud cover. the balloon, with capsule attached, 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

18、airliners (5.6 miles/9.17 kilometers) and into the stratosphere. 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 o

19、lympic diving platform.they blew it in 2008 when they 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

20、josh charnley in the opening game of the tournament.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 kick-chase. theyve been great at al