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高三数学第一轮复习教案第26课时—两角和与差的三角函数

▁▂▃▄▅▆▇█▉▊▋▌精诚凝聚 =^_^= 成就梦想 ▁▂▃▄▅▆▇█▉▊▋▌

一.课题:两角和与差的三角函数

二.教学目标:掌握两角和与差的三角函数公式,掌握二倍角公式;能运用这些公式进行

三角化简,求值等有关运算问题.

三.教学重点:公式的灵活运用.

四.教学过程:

(一)主要知识:

1.两角和与差的三角函数公式;二倍角公式;

2.降次公式: cos2 ? ? 1? cos 2? , sin2 ? ? 1? cos 2? .

2

2

(二)主要方法: 1.寻求所求结论中的角与已知条件中的角的关系,把握式子的变形方向,准确运用公
式; 2.三角变换主要体现在:函数名称的变换、角的变换、1的变换、和积的变换、幂的
变换等方面; 3.掌握基本技巧:切割化弦,异名化同名,异角化同角等.

(三)例题分析:

例 1.已知 cos? ? 1 , cos(? ? ? ) ? ? 11 ,? ? (0, ? ) ,? ? ? ? (? ,? ) 求 ? 的值.

7

14

2

2

解:∵ cos? ? 1 ,? ? (0, ? ) ,∴ sin? ? 4 3 ,

7

2

7

又∵ cos(? ? ? ) ? ? 11 ,? ? ? ? (? ,? ) ,∴ sin? ? 5 3 ,

14

2

14

∵ cos ? ? cos[(? ? ? ) ?? ] ? cos(? ? ? ) cos? ? sin(? ? ? ) sin? ? 1 , 2

又∵? ? (0, ? ) ,? ? ? ? (? ,? ) , ? ? (0,? ) ,

2

2

∴? ?? . 3

例 2.已知 A 为一三角形的內角,求 y ? cos2 A ? cos2 ( 2? ? A) 的取值范围. 3

解: cos2

A ? cos2 ( 2?

?

A)

?

1? cos 2A

1? cos 2( 2?

?

3

?

A)

3

2

2

? 1? cos 2A ? cos 4? cos 2A ? sin 4? sin 2 A

3

3

? 1? 1 cos 2A ? 3 sin 2A ? 1? cos(2 A ? ? ) .

2

2

3

∵ A 为一三角形內角, ? 1 ? cos(2A ? ? ) ? 1 ,

2

3

∴ y ? cos2 A ? cos2 ( 2? ? A) 的取值范围是 (1 ,1] .

3

2

例 3.求值: 2sin 50 ? sin 80 (1? 3 tan10 ) . 1? cos10

▃ ▄ ▅ ▆ ▇ █ █ ■ ▓点亮心灯 ~~~///(^v^)\\\~~~ 照亮人生 ▃ ▄ ▅ ▆ ▇ █ █ ■ ▓

▁▂▃▄▅▆▇█▉▊▋▌精诚凝聚 =^_^= 成就梦想 ▁▂▃▄▅▆▇█▉▊▋▌

解:原式

2sin 50 ? 2sin 80 (1 cos10 ? 3 sin10 ) 2sin 50 ? 2sin 80 cos(60 ?10 )

?

cos10 2

2

?

cos10

2 cos 5

2 cos 5

2( ?

2 sin 50 2

?

2 cos 50 ) 2

? 2 cos(50

? 45 ) ? 2 .

cos 5

cos 5

例 4.是否存在两个锐角? , ? 满足(1)? ? 2? ? 2? ;(2) tan ? ? tan ? ? 2 ? 3 同

3

2

时成立,若存在,求出? , ? 的值;若不存在,说明理由.

解:由(1)得 ? ? ? ? ? ,∴

2

3

3

?

tan(? 2

?

?)

?

tan ? 2
1? tan

? tan ? ? tan ?

,∴

2

???tan

? 2

? ???tan

? 2

? tan ? ? 2 ? ? tan ? ? 3 ?

3

,∴

??tan ?

? 2

3 ??tan ?

? 2? ?1

3



??tan ? ??tan

? ? 2

? ?

2 1

?

3 (∵ 0 ?

?

?

?

,∴

24

tan ? ? 1,舍去), 2



???? ? ????

? ?

? 6 ? 4

为所求满足条件的两个锐角.

(四)巩固练习: 1.化简 1? tan15 等于 1? tan15 ( A)
( A) 3

3 (B)
2

(C) 3

(D) 1

2.已知 sin? ? 2 cos? ? 0 ,则 sin 2? ? cos 2? ? ? 7 . 5

3.在

?ABC

中,

(1 ?

cot

A)(1 ?

cot

B)

?

2

,则

log2

sin

C

?

?

1 2



五.课后作业:《高考 A 计划》考点 26,智能训练 4,5,6,10,11,12,13,14.

▃ ▄ ▅ ▆ ▇ █ █ ■ ▓点亮心灯 ~~~///(^v^)\\\~~~ 照亮人生 ▃ ▄ ▅ ▆ ▇ █ █ ■ ▓


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