三角函数讲义5.doc

第5讲 三角函数的图象与性质1、考纲点击（1）能画出y=sinx, y=cosx, y=tanx的图象，了解三角函数的周期性；（2）理解正弦函数、余弦函数在区间0，2上的性质（如单调性、最大值和最小值以及与x轴的交点等），理解正切函数在区间内的单调性。2、热点提示（1）三角函数的值域、最值、单调性、周期性等性质是高考考查的重点；（2）三角函数的对称性也是高考的一个热点；（3）主要以选择题、填空题的形式考查。知识整合：1、周期函数（1）周期函数的定义对于函数f(x)，如果存在一个非零常数T，使得当x取定义域内的每一个值时，都有f(x+T)=f(x),那么函数f(x)就叫做周期函数。非零常数T叫做这个函数的周期。（2）最小正周期如果在周期函数f(x)的所有周期中存在一个最小的正数，那么这个最小正数就叫做f(x)的最小正周期。注：如果函数y=f(x)的周期是T，则函数y=f(x)周期是，而不是。2、正弦函数、余弦函数、正切函数的图象和性质函数y=sinxy=cosxy=tanx图象定义域xRxRxR且值域y|-1y1y|-1y1R单调性最值无最值奇偶性奇偶奇对称性对称中心（k,0）,kZ（k+,0）,kZ（,0）,kZ对称轴x= k+,kZx= k,kZ无周期22注：y=sinx与y=cosx的对称轴方程中的x都是它们取得最大值或最小值时相应的x，对称中心的横坐标都是它们的零点。典例分析：1、与三角函数有关的函数的定义域例1例1.求下列函数的定义域：（1）；（2）解：（1）即，故函数的定义域为且（2）即故函数的定义域为待添加的隐藏文字内容2点评：由几个函数的和构成的函数，其定义域是每一个函数定义域的交集；第（2）问可用数轴取交集2、三角函数单调区间的求法例2求下列函数的单调减区间：（1）； （2）；解：（1）因为，故原函数的单调减区间为（2）由，得，又，所以该函数递减区间为，即点评：利用复合函数求单调区间应注意定义域的限制3、三角函数的最小正周期例3求下列函数的最小正周期：（1）；（2） 解：（1）由函数的最小正周期为，得的周期（2） 4、三角函数的值域与最值例4（1）已知，求的最大值与最小值（2）求函数的最大值（3）求函数的最小值解：（1）由已知得：，则，当时，有最小值；当时，有最小值（2）设，则，则，当时，有最大值为（3）解法一：原式可化为，得，即，故，解得或（舍），所以的最小值为解法二：表示的是点与连线的斜率，其中点B在左半圆上，由图像知，当AB与半圆相切时，最小，此时，所以的最小值为点评：第（1）小题利用消元法，第（2）小题利用换元法最终都转化为二次函数求最值问题；但要注意变量的取值范围（3）解法一利用三角函数的有界性求解；解法二从结构出发利用斜率公式，结合图像求解例5.已知函数，（I）求的最大值和最小值； （II）若不等式在上恒成立，求实数的取值范围分析：观察角，单角二次型，降次整理为形式 解：（） 又，即，（），且，即的取值范围是Winger Tuivasa-Sheck, 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 not conceded a try since Josh Charnley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's 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 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."I'd give every award 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 Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League match 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 it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement the Russian side added: "We found no racist insults from 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 limited 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 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 Olympic 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 England's 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. They've 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-back 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.