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jian519:高中数学 纠错练习 三角函数与平面向量(2)

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纠错练习——三角函数与平面向量(2)
1 . 已 知 ( ) 4 A.- 3 5 B. 4
2

tan

θ

= 2 , 则

sin θ

2

+ sin

θ cos

θ

- 2cos θ

2



3 C.- 4

4 D. 5

2.(2010·威海模拟)已知方程 x +4ax+3a+1=0(a>0)的两根为 tan α 、tan β ,且 α 、β ∈

?-π ,π ?,则 tan α +β 的值是 ? 2 2? 2 ? ?
1 A. 2 B.-2 4 C. 3 1 D. 或-2 2

(

)

→ → → 3. △ABC 的三个内角 A,B,C 成等差数列,且(AB+AC)·BC=0,则△ABC 一定是( A.等腰直角三角形 C.等边三角形 B.非等腰直角三角形 D.钝角三角形

)

4.(2010·山东卷)定义平面向量之间的一种运算“⊙”,如下:对任意的 a=(m,n),

b=(p,q),令 a⊙b=mq-np.下面说法错误的是
A.若 a 与 b 共线,则 a⊙b=0 B.a⊙b=b⊙a

(

)

C.对任意的 λ ∈R,有(λ a)⊙b=λ (a⊙b)

D.(a⊙b) +(a·b) =|a| |b|

2

2

2

2

5.(2010·北京卷)某班设计了一个八边形的班徽(如图),它由腰长为 1,顶角为 α 的四 个 等腰三角形,及其底边构成的正方形所组成.该八边形的面积为 A.2sin α -2cos α +2 +3 C.3sin α - 3cos α +1 +1 D.2sin α -cos α B.sin α - 3cos α ( )

6.已知向量 a=(1,2),b=(1,1)且 a 与 a+λ b 的夹角为锐角,则实数 λ 的取值范围是 ________. 7.(2010·江苏卷)在锐角△ABC 中,角 A、B、C 的对边分别为 a、b、c.若 + =6cos C,

b a a b



tan C tan C + 的值是________. tan A tan B

8.已知|a|=4,|b|=3,(2a-3b)·(2a+b)=61.则 a 与 b 的夹角为________;|a+

b|=________. ;

9. 设函数 f(x)= 3cos x+sin xcos x-

2

3 . 2

(1)求函数 f(x)的最小正周期 T,并求出函数 f(x)的单调递增区间; (2)求在[0,3π )内使 f(x)取得最大值的所有 x 的和.

10. 设 A、B 分别是直线 y=

2 5 2 5 → x 和 y=- x 上的两个动点,且|AB|= 5 5

→ → → 20,动点 P 满足OP=O A +O B ,记动点 P 的轨迹为 C. (1)求轨迹 C 的方程; → → (2)若 D(0,16) ,M、N 是曲线 C 上的两个动点,且DM=λ DN(λ ≠1), 求实数 λ 的取值范围.

sin θ +sin θ cos θ -2cos θ 2 2 1 解析:sin θ +sin θ ·cos θ -2cos θ = = 2 2 sin θ +cos θ tan θ +tan θ -2 4+2-2 4 ,又 tan θ =2,故原式= = . 2 tan θ +1 4+1 5 答案:D 2 解析:∵a>0,∴tanα +tan β =-4a<0,tan α ·tan β =3a+1>0,又∵α 、
2

2

2

? π π? β ∈?- , ?, ? 2 2?
α +β ? π ? ? π ? ∴ α 、 β ∈ ?- ,0? , 则 ∈ ?- ,0? , ∴ tan(α + β ) = 2 ? 2 ? ? 2 ? tan α +tan β -4a = 1-tan α ·tan β 1- a+ α +β 2tan 2 4 4 = , ∴tan(α +β )= = , 整理得 3 3 2α +β 1-tan 2

2tan

2

α +β α +β α +β +3tan -2=0,解得 tan 2 2 2

1 =-2 或 (舍去).故选 B. 2 答案:B 3 解析:由题意可知,△ABC 中,BC 边上的中线又是 BC 边上的高,因此△ABC 是等腰 三角形,而三个内角 A,B,C 成等差数列,故角 B 为 60°,所以△ABC 一定是 等边三角形.答案:C 4 解析:选项 A,a 与 b 共线,则? λ 使得 a=λ b,则有 m=λ p,n=λ q,a⊙b=0;

选项 B,b⊙a=(p,q)⊙(m,n)=np-mq=-(a⊙b); 选项 C,λ a⊙b=(λ m,λ n)⊙(p,q)=λ mq-λ np=λ (a⊙b); 选项 D,(a⊙b) +(a·b) =(mq-np) +(mp+nq) =m q +n p +m p +n q =(m +n )(p +q )=|a| |b| . 答案:B 1 1 5 解析:等腰三角形的面积为 ×1×1·sin α = sin α , 2 2 等腰三角形的底边长为 a= 1 +1 -2×1×1×cos α 1 2 = 2-2cos α ,所以八边形面积为:4× sin α +a 2 =2sin α +2-2cos α . 答案:A 6 解析: ∵a 与 a+λ b 均不是零向量, 且其夹角为锐角, ∴a·(a+λ b)>0, 即 5+3λ >0, 5 ∴λ >- . 3 当 a 与 a+λ b 共线时,可设 a+λ b=ma(m∈R),即(1+λ ,2+λ )=m(1,2),
?1+λ =m ? ∴? ?2+λ =2m ?
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

,解得 λ =0,即当 λ =0 时,a 与 a+λ b 共线,

? 5 ? ∴λ ≠0.故 λ 的取值范围为?- ,0?∪(0,+∞). ? 3 ? ? 5 ? 答案:?- ,0?∪(0,+∞) ? 3 ?
7 解析:∵ + =6cos C,由余弦定理得 tan C tan C sin ∴ + = tan A tan B cos sin C sin C = · cos C sin Asin B 2c = = a2+b2-c2 a2+b2-c2 ab· 2ab

b a a b

a2+b2 a2+b2-c2 3 2 2 2 =6· ,∴a +b = c , ab 2ab 2

C?cos A cos B? + ? ? C?sin A sin B?

c2

2

2c = =4. 3 2 2 c -c 2 答案:4 8 解:(1)由(2a-3b)·(2a+b)=61, 得 4|a| -4a·b-3|b| =61, ∵|a|=4,|b|=3,代入上式得 a·b=-6,
2 2

2

a·b -6 1 ∴cos θ = = =- . |a||b| 4×3 2
又 0°≤θ ≤180°,∴θ =120°. (2)|a+b| =(a+b) =|a| +2a·b+|b|
2 2 2 2 2 2

=4 +2×(-6)+3 =13,∴|a+b|= 13.

9 审题提示:首先化 f(x)为 Asin(ω x+φ )的形式,再由三角函数性质求. 解:(1)f(x)= 3 1 3 (1+cos 2x)+ sin 2x- 2 2 2

π? 1 3 ? = sin 2x+ cos 2x=sin?2x+ ?. 3? 2 2 ? 故 T=π . 由 2kπ - ∴kπ - π π π ≤2x+ ≤2kπ + (k∈Z),得 2 3 2

5π π ≤x≤kπ + , 12 12

函数 f(x)的单调递增区间为?kπ -

? ?

5 π π ,kπ + ? (k∈Z). 12 12? ?

2 5 2 5 10 解:(1)设 P(x,y),因为 A、B 分别为直线 y= x 和 y=- x 上的 5 5 点,故可设 A?x1,

? ?

2 5 ? 2 5 ? ? x1?,B?x2,- x2?. 5 ? 5 ? ?



→ → OP=O A +O B . x=x1+x2, ? ? ∴? 2 5 y= x1-x2 ? 5 ?
→ 又|AB|= 20, 4 5 2 4 2 2 2 ∴(x1-x2) + (x1+x2) =20.∴ y + x =20. 5 4 5 故 C 的方程为 + =1. 25 16 → → (2)设 N(s,t),M(x,y),则由DM=λ DN, 可得(x,y-16)=λ (s,t-16). 故 x=λ s,y=16+λ (t-16). 又 M、N 在曲线 C 上,

x1+x2=x, ? ? ∴? 5y x1-x2= . ? 2 ?

x2

y2

? ?25 16=1, ?λ s +λ t-16λ + ? ? 25 16

2 2

s2

t2

2

由题意,λ ≠0,且 λ ≠1. =1.

解得 t=

17λ -15 ?17λ -15?≤4,得3≤λ ≤5(λ ≠1). .又|t|≤4,∴? ? 2λ 5 3 ? 2λ ?


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