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guzhijia.com:数学:1.4.1《正弦、余弦函数图象》课件(新人教A版必修4)_图文

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1.4 三角函数的图象与性质

1.4.1正弦函数、余弦函数的图象

问题提出

t

? 1 ? 5730 p ? ? ? ?2?

1.在单位圆中,角α 的正弦线、余弦线 分别是什么?
y

sinα =MP cosα =OM
O M

P(x,y)

x

2.任意给定一个实数x,对应的正弦值 (sinx)、余弦值(cosx)是否存在?惟一?

3.设实数x对应的角的正弦值为y,则对 应关系y=sinx就是一个函数,称为正弦 函数;同样y= cosx也是一个函数,称为 余弦函数,这两个函数的定义域是什么?

4.一个函数总具有许多基本性质,要直 观、全面了解正、余弦函数的基本特性, 我们应从哪个方面人手?

知识探究(一):正弦函数的图象

思考1:作函数图象最原始的方法是什么?

思考2:用描点法作正弦函数y=sinx在[0, 2π ]内的图象,可取哪些点? 思考3:如何在直角坐标系中比较精确地 描出这些点,并画出y=sinx在[0,2π ] 内的图象?

y 1

y ? sin x, x[0, 2??

π O -1
p 2

3p 2

2π x

思考4:观察函数y=sinx在[0,2π ]内的 图象,其形状、位置、凸向等有何变化 规律?

思考5:在函数y=sinx,x∈[0,2π ]的 图象上,起关键作用的点有哪几个?
y 1
3p

π
O -1
p 2

2

2π x

思 考 6 : 当 x∈[2π , 4π ], [-2π , 0],?时,y=sinx的图象如何?
y
1 -6π -4π -5π -3π -1 -2π -π
O

π





5π 6π x

思考7:函数y=sinx,x∈R的图象叫做正 弦曲线,正弦曲线的分布有什么特点?
y 1 -6π -4π -5π -3π -1 -2π -π
O

π 2π

3π 4π

5π 6π x

思考8:你能画出函数y=|sinx|, x∈[0,2π ]的图象吗?
y 1

O -1

π



x

知识探究(二):余弦函数的图象

思考1:观察函数y=x2与y=(x+1)2 的图 象,你能发现这两个函数的图象有什么 内在联系吗?
y

-1

o

x

思考2:一般地,函数y=f(x+a)(a>0) 的图象是由函数y=f(x)的图象经过怎样 的变换而得到的? 向左平移a个单位. 思考3:设想由正弦函数的图象作出余 弦函数的图象,那么先要将余弦函数 y=cosx转化为正弦函数,你可以根据哪 个公式完成这个转化?

思考4:由诱导公式可知,y=cosx与 p y = sin( + x ) 是同一个函数,如何作函 2 p 数 y = sin( 2 + x )在[0,2π ]内的图象?
y 1

y=sinx
?? 2

?

? 2

O -1

π



x

思考5:函数y=cosx,x∈[0,2π ]的图 象如何?其中起关键作用的点有哪几个?
y 1

O -1

? 2

π

?? 2



x

思考6:函数y=cosx,x∈R的图象叫做余 弦曲线,怎样画出余弦曲线,余弦曲线 的分布有什么特点?
? ?? 2

?
?? 2

?? 2

?
?? 2

? 2

1 O

y ?

?? 2
?? 2
?? 2

?? 2

2

x
??? 2

?

??? 2

?

?

-1

理论迁移

例1 用“五点法”画出下列函数的 简图: (1)y=1+sinx,x∈[0,2π ]; (2)y=-cosx,x∈[0,2π ] .

x sinx

p

3p

0
0

2

p

2

2p

1

0

-1

0

1+sinx
y 2 1

1

2

1

0

1

y=1+sinx
3p

π O -1
p 2

2

2π x

x cosx -cosx
y

p

3p

0 1 -1

2

p

2

2p

0 0

-1 1

0 0

1 -1

y=-cosx
1 O -1
p 2
3p 2

2π x

π

例2 当x∈[0,2π ]时,求不等式
cos x ? 1 2
1
y = 1 2

的解集.
y

O -1

? 2

π

?? 2



x

[0,

p 3

]U[

5p 3

, 2p ]

小结作业

1.正、余弦函数的图象每相隔2π 个单位 重复出现,因此,只要记住它们在[0, 2π ]内的图象形态,就可以画出正弦曲 线和余弦曲线. 2.作与正、余弦函数有关的函数图象, 是解题的基本要求,用“五点法”作图 是常用的方法.

3.正、余弦函数的图象不仅是进一步研 究函数性质的基础,也是解决有关三角 函数问题的工具,这是一种数形结合的 数学思想.

作业:P34练习:2 P46习题1.4 A组: 1


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