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复习

2、在三角形ABC中, tanAtanB>1,则ABC一定是 ( )三角形
.

2 2 2 sin(? ? ? ) sin ? cos ? ? cos ? sin ? tan(? ? ? ) ? ? cos(? ? ? ) cos ? cos ? ? sin ? sin ?

当? ?

?

,? ?

?

,? ? ? ?

?



sin ? cos ? cos ? sin ? ? tan ? ? tan ? cos ? cos ? cos ? cos ? ? ? sin ? sin ? 1 ? tan ? tan ? 两角和与差的正切公式 1? cos ? cos ?

tan ? ? tan ? tan(? ? ? ) ? 1 ? tan ? tan ?

(? ?

?
2

,? ?

?
2
.

,? ? ? ?

?
2

)

例1、不查表求值。

1)tan105°
2)tan75°

3)tan15°
.

※ 2α-β= α+(α-β)

※ 2α=(α-β)+(α+β)
.

.

例5、计算

.

变形公式
tanα+tanβ= tan(α+β)·(1-tanαtanβ) tanα-tanβ= tan(α-β)·(1+tanαtanβ)

.

例2、求证tan3α-tan2αtanα=tan3αtan2αtanα
=tan(3α-2α)(1+tan3αtan2α)-tanα

.

例3、在非直角三角形中,求证: tanA+tanB+tanC=tanAtanBtanC A+B+C=π =tan(A+B)(1-tanAtanB)+tanC =tan(π-C)(1-tanAtanB)+tanC

=1+tan(α+β)(1-tanαtanβ)+tanαtanβ
.

§4.6两角和与差的正弦、余弦、正切(3)

两角和与差的正弦、余弦、正切公式的内在联系
-?代?
?
? ?? ? ? ?

S(?+?)

2

C(?+?)

-?代?

?

C(?-?)

2

? ?? ? ? ?

S(?-?)

相除
S ?? ? ? ? C?? ? ? ?
-?代?

相除
S ?? ? ? ? C?? ? ? ?

T(?+?)

T(?-?)

.

思考

比较sin2α与sinα·cosα的 值,猜想sin2α的公式
sin(α+α)=? cos2α? tan2α?
.

sin2α=2sinαcosα

1 ? cos 2? 1.降幂公式: cos ? ? 2
2

1 ? cos 2? sin ? ? 2
2

2 1 ? cos 2 ? ? 2cos ? 2.升幂公式:
.

1 ? cos 2? ? 2sin 2 ?

5 ? sin ? ? , ? ? ( ,? ) 已知 13 2

,(1)求sin(? ? 4 ), cos(? ? 6 ), tan(? ? 3 )

?

?

?

的值;(2)求 sin 2? , cos 2? , tan 2? 的值。

(1)

(2)

.

例2、求下列各式的值。
1) sin 67 30 cos67 30 12 0 2 tan150 3) 2 0 1 ? tan 150
.

0

'

0

'

2)2 cos

2

?

?1

.

(1)由-2cosα +sinα =0即sinα =2cosα .
4 sin 2 ? ?又∵ . α ∈(π, 5 2 5 5 ? sin? ? ? ,? cos? ? ? , 5 5 ? ? ? ? sin(? ? ) ? sin?cos ? cos?sin 4 4 4 2 ? (sin? ? cos?) 2 2 2 5 5 3 10 ? (? ? )?? . 2 5 5 10

又sin2α +cos2α =1得

),

3? 2

(2)由(1)可得tanα =2,
? 1 ? tan? 1 ? 2 ? tan(? ? ) ? ? ? ?3. 4 1 ? tan? 1 ? 2
.

(09茂名市模拟)已知函数f(x)=cos2x+sinxcosx(x∈R) 3π (Ⅰ)求f( )的值; 8 (Ⅱ)求f(x)的单调递增区间.
解:

1 ? cos 2 x 1 f ( x) ? ? sin 2 x 2 2

2 2 2 1 ? ( sin 2 x ? cos 2 x) ? 2 2 2 2 2 ? 1 ? sin(2 x ? ) ? 2 4 2

3? 2 1 1 (I) f ( ) ? sin ? ? ? 8 2 2 2

.

(II)令2k? ?

?
2

? 2x ?

?
4

? 2 k? ?

?
2

,k ?Z

3? ? ? 2k? ? ? 2 x ? 2 k? ? , k ? Z 4 4
3? ? ? x ? k? ? (k ? Z ) 即 k? ? 8 8
时,f(x)单调递增.
] (k ? Z )

3? 3? ∴f(x)单调递增区间为[ k? ? , k? ? 8 8

.

π? ? 1 ? 2 cos ? 2 x ? ? 4? ? f ( x ) ? (07重庆文)已知函数 π? ? sin ? x ? ? 2? ?

(Ⅰ)求 f ( x) 的定义域; (Ⅱ)若角a在第一象限且

3 cos ? ? 5

,求 f (? )

π? ? sin ? x ? ? ? 0 解:(Ⅰ) 由 2? ? π x ? kπ ? (k ? Z) 即

π 得 x ? ? ? kπ , 2

2

π ? ? 故 f ( x) 的定义域为 ? x ? R | x ? kπ ? ,k ? Z ? 2 ? ?
.

4 ?3? 2 (Ⅱ)由已知条件得 sin ? ? 1 ? cos ? ? 1 ? ? ? ? 5 ?5?

2

π? ? 1 ? 2 cos ? 2? ? ? 4? ? f ( ? ) ? 从而 π? ? sin ? ? ? ? 2? ?

π π? ? 1 ? 2 ? cos 2? cos ? sin 2? sin ? 4 4? ? ? cos ?

1 ? cos 2? ? sin 2? 2cos 2 ? ? 2sin ? cos ? ? ? cos ? cos ?

14 ? 2(cos ? ? sin ? ). ? 5


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