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好属日视频84gao com:高中数学 第一章 三角函数 1.4.3 正切函数的性质与图象课件4 新人教A版必修4

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1.4.3正切函数的性质与图象
1

三角函数线

y

?
PT

A(1,0) o M x MP是正弦线

y

OM是余弦线

T

AT是正切线

Mo

Ax

P

?

y

P

Mo

y

o

Ax T
MAx PT

2

现在利用正切线画出函数y ? tan x, x ?(? ? , ? )的图象
22
y
?

?1

?

?

o1

?? 2

?? 4

?

0?
4

?x
2

?

? ?1

?

3

利用正切函数的周期性,把图象向左,右扩展,得到正切函数 y ? tan x, x ? R且x ? ? ? k? , (k ? Z )的图象,并把它叫做正切曲线.
2 y

? 3? ??

??

2

2

0

?

?

2

3? 2

x

4

正切函数图象的简单画法: 三点两线法.

“三点”: (0,0)、(? ,1)、?? ? ? ,?1?? 4 ?4 ?

“两线”:x

?

?
2

和x

?

?

?
2

y

1
?? 4

? 3? ??
2

?? 2

0? ?

?

42

-1

3? 2

x

5

y

? 3? ??

??

0

?

?

2

2

2

3? 2

x

结合正切函数图像研究正切函数的性质:定义域、值域、周期性、 奇偶性和单调性.

6

例1.求函数y ? tan(x ? ? )的定义域.
4
7

正切函数的图像和性质

例2.不通过求值,比较下列各组中两个正切函数值的大小:

(1)tan167?



tan

173?

;(2)tan?? ?

?

11?
4

??与 ?

tan?? ?

?

13?
5

?? ?



8

练习:

1.观察正切曲线,写出满足下列条件的x的

值的范围. (1) tan x ? 0,(2) tan x ? 0,(3) tan x ? 0

解:

y

? 3? ??
2

?? 2

0

? 2

?

3? 2

x

(1).x ?(k? ? ? , k? ),(k ? Z)
2

(2).x ? k? ,

(k ? Z)

(3).x ?(k? , k? ? ? ),(k ? Z)
2

9

例3 求下列函数的单调区间:
(1) y ? 3tan(1 x ? ? );
24
10

(2)变题y ? 3tan(? 1 x ? ? )的单调递减区间.
24
11


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