第2讲 同角三角函数的基本关系与诱导公式
12
1.若cos x=,且x为第四象限的角,则tan x的值为________.
13
5-135
解析:因为x为第四象限的角,所以sin x=-1-cos2x=-,于是tan x==
1312
135-. 12
5
答案:-
122.已知sin?解析:sin?1答案: 5
3π?π?3.已知cos?-φ?=,且|φ|<,则tan φ=________.
2?2?23?π?解析:cos?-φ?=sin φ=, 2?2?
π1
又|φ|<,则cos φ=,所以tan φ=3.
22答案:3
cos(α-π)π3π
4.化简:·sin(α-)·cos(-α)=________.
sin(π-α)22
cos(α-π)π3π-cos α
解析:·sin(α-)·cos(-α)=·(-cos α)·(-sin
sin(π-α)22sin αα)=-cosα.
答案:-cosα
5.如果f(tan x)=sinx-5sin x·cos x, 那么f(5)=________. 解析:f(tan x)=sinx-5sin x·cos x sin xsin x2-52
cos xsinx-5sin x·cos xcosx== 222
sinx+cosxsinx2+1cosx
1
2
22
2
2
?5π+α?2?=1,那么cos α=________. ?5?
?5π+α?=sin?π+α?=cos α=1.
??2?5?2???
tanx-5tan x5-5×5=,所以f(5)==0. 22
tanx+15+1答案:0
1?ππ??3?6.已知sin θ=-,θ∈?-,?,则sin(θ-5π)·sin?π-θ?的值是
3?22??2?________.
1?ππ?解析:因为sin θ=-,θ∈?-,?, 3?22?
222
所以cos θ=1-sinθ=.所以原式=-sin(π-θ)·(-cos θ)=sin θcos
312222θ=-×=-.
339
22答案:- 9
7.(2019·江苏省四校联考)已知sin x=则tan x=________.
22
m-34-2m?3π?,cos x=,且x∈?,2π?,m+5m+5?2?
?m-3?+?4-2m?=1,?3π?解析:由sinx+cosx=1,即?得m=0或m=8.又x∈?,2π?,???
?m+5??m+5??2?
2
2
22
343
所以sin x<0,cos x>0,所以当m=0时,sin x=-,cos x=,此时tan x=-;当m554512
=8时,sin x=,cos x=-(舍去).
1313
3
综上知,tan x=-.
43
答案:-
48.若f(α)=________.
解析:①当k为偶数时,设k=2n(n∈Z),
sin(2nπ+π+α)·cos(2nπ+π-α)原式= sin(-α)·cos α=
sin(π+α)·cos(π-α)
=-1;
-sin α·cos α
sin[(k+1)π+α]·cos[(k+1)π-α]
(k∈Z),则f(2 018)=
sin(kπ-α)·cos(kπ+α)
②当k为奇数时,设k=2n+1(n∈Z),
sin[(2n+2)π+α]·cos[(2n+2)π-α]原式= sin[(2n+1)π-α]·cos[(2n+1)π+α]
2
=
sin α·cos(-α)
=-1.
sin(π-α)·cos(π+α)
综上所述,当k∈Z时,f(α)=-1,故f(2 018)=-1. 答案:-1
45?4?9.sinπ·cos π·tan?-π?的值是________. 36?3?π?π?π????解析:原式=sin?π+?·cos?π-?·tan?-π-?
3?6?3????π??π??π??=?-sin ?·?-cos ?·?-tan ?
3??6??3??=?-?
?333??3?
?×?-?×(-3)=-4. 2??2?
33
答案:- 4
πcosx10.当0
4cos xsin x-sinxπ
解析:当0
4
cosx1
f(x)=2=2,
cos xsin x-sinxtan x-tanx设t=tan x,则0
111
≥=4.当且仅当t=1-t,2=t-tt(1-t)t+(1-t)2
????2??
2
2
1
即t=时等号成立.
2
答案:4 11.化简:
???
1+sin α
-1-sin α
1-sin α??
?·?1+sin α??
2
1+cos α
-1-cos α
21-cos α?
?.
1+cos α?
?
解:原式=?
????
=?
(1+sin α)
-2
cosα
2
(1-sin α)?
?· 2
cosα?
2
(1+cos α)
-2
sinα
(1-cos α)?
? 2
sinα?
?1+sin α-1-sin α?·?1+cos α-1-cos α? ????|cos α||cos α|??|sin α||sin α|?
??4,α在第一、三象限时,2sin α2cos α
=·=? |cos α||sin α|??-4,α在第二、四象限时.
3
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