三角函数的概念及公式

三角函数概念及性质

任意角、三角函数的概念

1.(1)已知α是第二象限角,那么α2(  )                                                                                   A.第一象限角  B.第二象限角  C.第一或第二象限角  D.第一或第三象限角                     1.(1)已知 \alpha 是第二象限角, 那么 \frac{\alpha}{2} 是(~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ A.第一象限角~~ B.第二象限角~~ C.第一或第二象限角~~ D.第一或第三象限角~~~~~~~~~~~~~~~~~~~~~
(2)已知α是第二象限角,那么α3(  )                                                                                       A.第二或第三或第四象限角  B.第一或第二或第三象限角                                                  C.第一或第二或第四象限角  D.第一或第三或第四象限角                                                 (2)已知 \alpha 是第二象限角, 那么 \frac{\alpha}{3} 是(~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ A.第二或第三或第四象限角~~ B.第一或第二或第三象限角~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ C.第一或第二或第四象限角~~ D.第一或第三或第四象限角~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:(1)D,(2)C;提示:α(π2+2kπ,π+2kπ),kZ,α2(π4+kπ,π2+kπ), kZ,在一、三象限;α3(π6+2kπ3,π3+2kπ3),kZ,在第一或第二或第四象限.答案:(1)D,(2)C;提示:\alpha \in \left ( \frac{\pi }{2}+2k\pi , \pi +2k \pi \right ) ,k\in{Z},则\frac{\alpha }{2} \in \left ( \frac{\pi }{4}+k\pi , \frac{\pi }{2} +k \pi \right ) ,~\\k\in{Z},在一、三象限;\frac{\alpha }{3} \in \left ( \frac{\pi }{6}+\frac{2k\pi}{3} , \frac{\pi }{3} +\frac{2k\pi}{3} \right ) ,k\in{Z},在第一或第二或第四象限.

2.sinxcosx>0,sinx+cosx>0,x2(  )                                                                    A.第一象限角  B.第二象限角  C.第一或第二象限角  D.第一或第三象限角                     2.若\sin x \cos x>0,\sin x +\cos x>0,则\frac{x}{2}是(~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\A.第一象限角~~ B.第二象限角~~ C.第一或第二象限角~~ D.第一或第三象限角~~~~~~~~~~~~~~~~~~~~~
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答案:D;提示:x为第一象限角,原理同上                                                                         答案:D;提示:x为第一象限角,原理同上~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

3.化简1+sinα1sinα1sinα1+sinα,其中α为第二象限角.                                                       3.化简 \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}-\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}} , 其中 \alpha 为第二象限角.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:2tanα;提示:原式(1+sinα)21sin2α(1sinα)21sin2α                                        答案:-2\tan \alpha;提示:原式\Rightarrow \sqrt{\frac{(1+\sin \alpha)^2}{1-\sin^2 \alpha}}-\sqrt{\frac{(1-\sin \alpha)^2}{1-\sin^2 \alpha}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

4.已知角α的顶点为坐标原点,始边为x轴的非负半轴,若点P(sinα,tanα)在第四象限,   则角α的终边在(  )                                                                                                                    A.第一象限  B.第二象限  C.第三象限  D.第四象限                                                             4.已知角 \alpha 的顶点为坐标原点, 始边为 x 轴的非负半轴, 若点 P(\sin \alpha , \boldsymbol{\operatorname { t a n }} \alpha )在第四象限, ~~~\\则角 \alpha 的终边在( ~~)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ A. 第一象限~~ B. 第二象限~~ C. 第三象限~~ D. 第四象限~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:B;提示:P(sinα,tanα)在第四象限,sinα>0tanα<0,α在第二象限.       答案:B;提示: P(\sin \alpha , \boldsymbol{\operatorname { t a n }} \alpha )在第四象限,\sin \alpha >0且 \tan {\alpha}<0,\therefore \alpha在第二象限.~~~~~~~

5.已知角α的终边上有一点P的坐标是(3a,4a),其中a0,sinα,cosα,tanα的值.      5.已知角 \alpha 的终边上有一点 P 的坐标是 (3 a, 4 a) , 其中 a \neq 0 ,求 \sin \alpha, \cos \alpha, \tan \alpha 的值.~~~~~~
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答案:a>0,可令a=1,sinα=45,cosα=35,tanα=43;                                                  a<0,可令a=1,sinα=45,cosα=35,tanα=43.                                                     答案:a>0, 可令a=1,\sin \alpha=\frac{4}{5}, \cos \alpha=\frac{3}{5}, \tan \alpha=\frac{4}{3};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\a<0, 可令a=-1,\sin \alpha=-\frac{4}{5}, \cos \alpha=-\frac{3}{5}, \tan \alpha=\frac{4}{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

6.已知角α的终边过点P(8m,6sin30),cosα=45,m的值为(  )                       A.12  B.32  C.12  D.32                                                                                                 6. 已知角 \alpha 的终边过点 P\left(-8 m,-6 \sin 30^{\circ}\right) , 且 \cos \alpha=-\frac{4}{5} , 则 m 的值为 ( ~~)~~~~~~~~~~~~~~~~~~~~~~~\\ A. -\frac{1}{2} ~~ B. -\frac{\sqrt{3}}{2} ~~ C. \frac{1}{2} ~~ D. \frac{\sqrt{3}}{2} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:C;提示:由题意得点P(8m,3),r=64m2+9所以cosα=8m64m2+9      =45,所以m>0,解得m=12.                                                                                               答案: C; 提示: 由题意得点 P(-8 m,-3) , r=\sqrt{64 m^{2}+9} 所以 \cos \alpha=\frac{-8 m}{\sqrt{64 m^{2}+9}}~~~~~~\\=-\frac{4}{5} , 所以 m>0 , 解得 m=\frac{1}{2} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

7.已知角α的始边与x轴非负半轴重合,终边在直线y=4x上,求sinα2cosαtanα的值.    7.已知角 \alpha 的始边与 x 轴非负半轴重合, 终边在直线 y=4 x 上,求 \frac{\sin \alpha-2 \cos \alpha}{\tan \alpha} 的值.~~~~\\
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答案:17341734;提示:在第一象限取点(1,4),cosα=117,sinα=417,       tanα=4;在第三象限取点(1,4),cosα=117,sinα=417,tanα=4.         ★注:由y=4x,tanα=4,sinαcosα不唯一.                                                                  答案:\frac{\sqrt{17}}{34} 或 -\frac{\sqrt{17}}{34} ;提示:在第一象限取点(1,4),得 \cos \alpha=\frac{1}{\sqrt{17}}, \sin \alpha=\frac{4}{\sqrt{17}}, ~~~~~~~\\\tan \alpha=4 ;在第三象限取点(-1,-4),得 \cos \alpha=-\frac{1}{\sqrt{17}}, \sin \alpha=-\frac{4}{\sqrt{17}}, \tan \alpha=4 .~~~~~~~~~\\★注:由y=4x,\tan \alpha=4,\sin \alpha和\cos \alpha不唯一.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

8.(1)已知sinα=35,cosα,tanα的值.                                                                             8.(1)已知 \sin \alpha=-\frac{3}{5} , 求 \cos \alpha, \tan \alpha 的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
(2)已知tanφ=3,sinφ,cosφ的值.                                                                             (2)已知 \tan \varphi=-\sqrt{3} , 求 \sin \varphi, \cos \varphi 的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(3)求值:cos225;sin11π3;sin(16π3);cos65π6.                                                             (3)求值:\cos 225^{\circ};\sin \frac{11 \pi}{3};\sin \left(-\frac{16 \pi}{3}\right);\cos \frac{65 \pi}{6}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:(1)α为第三象限角时,cosα=45,tanα=34;α为第四象限角时,                     cosα=45,tanα=34.                                                                                                            (2)α为第二象限角时,sinα=32,cosα=12;α为第四象限角时,                              sinα=32,cosα=12.★注:可令y=3,x=1,快速求sinxcosx,再加符号.(3)22;32;32;32.提示:根据对称性转化到第一象限角.                                    答案:(1)\alpha为第三象限角时, \cos \alpha=-\frac{4}{5}, \tan \alpha =\frac{3}{4};\alpha为第四象限角时,~~~~~~~~~~~~~~~~~~~~~\\ \cos \alpha=\frac{4}{5}, \tan \alpha =-\frac{3}{4}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\(2)\alpha为第二象限角时, \sin \alpha=\frac{\sqrt{3}}{2}, \cos \alpha =-\frac{1}{2};\alpha为第四象限角时,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \sin \alpha=-\frac{\sqrt{3}}{2}, \cos \alpha =\frac{1}{2}.★注:可令y=\sqrt{3},x=1,快速求|\sin x|和|\cos x|,再加符号.\\(3)-\frac{\sqrt{2} }{2} ;-\frac{\sqrt{3} }{2};\frac{\sqrt{3} }{2};-\frac{\sqrt{3} }{2}.提示:根据对称性转化到第一象限角.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

9.在平面直角坐标系中,动点A在以原点为圆心.1为半径的圆上,2rad/s的角速度按    逆时针方向做匀速圆周运动;动点B在以原点为圆心,2为半径的圆上,1rad/s的角速度按逆时针方向做匀速圆周运动.A,B分别以A0(0,1),B0(2,0)为起点同时开始运动,t s.动点A,B的坐标分别为(x1,y1),(x2,y2).y1+x2的最小值为()                      A.3B.2C.32D.1                                                                                         9.在平面直角坐标系中,动点 A 在以原点为圆心. 1 为半径的圆上,以 2 \mathrm{rad} / \mathrm{s} 的角速度按~~~~ \\逆时针方向做匀速圆周运动;动点 B 在以原点为圆心, 2 为半径的圆上,以 1 \mathrm{rad} / \mathrm{s} 的角速 \\度按逆时针方向做匀速圆周运动. A, B 分别以 A_{0}(0,1), B_{0}(2,0) 为起点同时开始运动,经\\过 t \mathrm{~s} 后.动点 A, B 的坐标分别为 \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) .则 y_{1}+x_{2} 的最小值为(\quad)~~~~~~~~~~~~~~~~~~~~~~\\ A.-3\quad B.-2\quad C. -\frac{3}{2} \quad D.-1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:C;提示:y1+x2=sin(π2+2t)+2cost.              答案:C;提示:y_1+x_2=\sin{(\frac{\pi}{2}+2t)}+2\cos t.~~~~~~~~~~~~~~

三角公式的应用

1.已知tanα=2,(1)sinα+cosαsinαcosα的值;(2)sinαcosα;(3)cos2α+sin2α.                1. 已知 \tan \alpha=2 , (1)求 \frac{\sin \alpha+\cos \alpha}{\sin \alpha-\cos \alpha} 的值;(2)求\sin \alpha\cos \alpha;(3)\cos ^2{\alpha+\sin 2\alpha.}~~~~~~~~~~~~~~~~
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答案:(1)3;(2)25;(3)1;提示:利用齐次式,转化为tanα.                                                  答案:(1)3;(2)\frac{2}{5};(3)1;提示:利用齐次式,转化为\tan \alpha.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

2.1tan(απ4)1+tan(απ4)=12,cos2α的值.                                                                               2.若 \frac{1-\tan \left(\alpha-\displaystyle \frac{\pi}{4}\right)}{1+\tan \left(\alpha-\displaystyle\frac{\pi}{4}\right)}=\displaystyle\frac{1}{2} ,求 \cos 2 \alpha 的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:35;提示:由已知tan(απ4)=13,tanα=2,cos2α=cos2αsin2α=1tan2α1+tan2α=35.                                                                                                                 答案: -\frac{3}{5} ; 提示: 由已知 \Rightarrow \tan \left(\alpha-\frac{\pi}{4}\right)=\frac{1}{3}, \Rightarrow \tan \alpha=2, \therefore \cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha\\= \frac{1-\tan ^{2} \alpha}{1+\tan ^{2} \alpha}=-\frac{3}{5} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

3.已知角α,β满足tanα=13,2sinβ=sin(2α+β),tanβ.                                              3.已知角 \alpha, \beta 满足 \tan \alpha=\frac{1}{3}, 2 \sin \beta=\sin (2 \alpha+\beta) , 求 \tan \beta.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:12;提示:tanα=13,sin2α=35,cos2α=45,原式 2sinβ=sin(2α+β)        展开代入得tanβ=12.                                                                                                             答案:\frac{1}{2} ;提示:\tan \alpha=\frac{1}{3}, \Rightarrow \sin2 \alpha=\frac{3}{5},\cos2 \alpha=\frac{4}{5},原式~2 \sin \beta=\sin (2 \alpha+\beta)~~~~~~~~\\ 展开代入得 \tan \beta=\frac{1}{2}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

4.α为锐角,tanα=1cos2α+1,tanα=()                                                                A.12  B.1  C.23  D.3                                                                                                      4.若 \alpha 为锐角, \tan \alpha=\frac{1}{\cos 2 \alpha+1} , 则 \tan \alpha=(\quad) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\A. \frac{1}{2} ~~ B. 1~~ C. 2-\sqrt{3} ~~ D. \sqrt{3} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:B;提示:tanα=sin2α+cos2α2cos2α=tan2α+12,tan2α2tanα+1=0,      解得tanα=1.                                                                                                                         答案 :B;提示: \tan \alpha=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{2 \cos ^{2} \alpha}=\frac{ \tan ^{2} \alpha+1}{2} ,即 \tan ^{2} \alpha-2 \tan \alpha+1=0 , ~~~~~~\\解得 \tan \alpha=1 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

5.已知θ(0,π),sinθ+cosθ=15,求值(1)sinθcosθ;(2)sinθ;(3)cosθ;(4)tanθ.       5.已知 \theta \in(0, \pi), \sin \theta+\cos \theta=\frac{1}{5} , 求值(1)\sin \theta-\cos \theta; (2)\sin \theta;(3)\cos \theta;(4)\tan \theta.~~~~~~~\\
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答案:(1)sinθcosθ=75;(2)sinθ=45;(3)cosθ=35;(4)tanθ=43.                       提示:由题意知sinθ+cosθ=15,平方得2sinθcosθ=2425<0,θ(0,π),         π2<θ<π,sinθcosθ>0,sinθcosθ=12sinθcosθ=75,              与已知联立,sinθ=45,cosθ=35.tanθ=43.                                                           答案:(1) \sin \theta-\cos \theta=\frac{7}{5};(2) \sin \theta=\frac{4}{5} ; (3)\cos \theta=-\frac{3}{5} ; (4)\tan \theta=-\frac{4}{3} .~~~~~~~~~~~~~~~~~~~~~~~\\提示:由题意知 \sin \theta+\cos \theta=\frac{1}{5}, 平方得 2 \sin \theta \cos \theta=-\frac{24}{25}<0 , 又 \because \theta \in(0, \pi), ~~~~~~~~~\\\therefore \frac{\pi}{2}< \theta<\pi, \quad \therefore \sin \theta-\cos \theta>0, \quad \therefore \sin \theta-\cos \theta=\sqrt{1-2 \sin \theta \cos \theta}=\frac{7}{5}, ~~~~~~~~~~~~~~\\与已知联立, \sin \theta=\frac{4}{5}, \cos \theta=-\frac{3}{5} . \therefore \tan \theta=-\frac{4}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

6.已知π<x<0,sin(π+x)cosx=15,sin2x+2sin2x1tanx.                                    6.已知 -\pi < x < 0, \sin (\pi+x)-\cos x=-\frac{1}{5} , 求 \frac{\sin 2 x+2 \sin ^{2} x}{1-\tan x}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:24175;提示:由已知sinx+cosx=15,2sinxcosx=2425,                 (sinxcosx)2=12sinxcosx=4925,缩小角的范围π2<x<0                      sinxcosx=75,sinx=35,cosx=45,tanx=34.                                               答案: -\frac{24}{175} ;提示: 由已知\Rightarrow \sin x+\cos x=\frac{1}{5}, \Rightarrow 2 \sin x \cos x=-\frac{24}{25} ,\Rightarrow~~~~~~~~~~~~~~~~~\\(\sin x-\cos x)^{2}=1-2 \sin x \cos x =\frac{49}{25} , 缩小角的范围\Rightarrow -\frac{\pi}{2}< x< 0 \Rightarrow ~~~~~~~~~~~~~~~~~~~~~~\\\sin x-\cos x=-\frac{7}{5} ,\Rightarrow\sin x =-\frac{3}{5},\cos x=\frac{4}{5},\tan x=-\frac{3}{4}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

7.化简sin(2πα)cos(π+α)cos(π2+α)cos(11π2α)cos(πα)sin(3πα)sin(πα)sin(9π2+α).                                              7.化简\frac{\sin (2 \pi-\alpha) \cos (\pi+\alpha) \cos \left(\displaystyle\frac{\pi}{2}+\alpha\right) \cos \left(\displaystyle\frac{11 \pi}{2}-\alpha\right)}{\cos (\pi-\alpha) \sin (3 \pi-\alpha) \sin (-\pi-\alpha) \sin \left(\displaystyle\frac{9 \pi}{2}+\alpha\right)}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:tanα;提示:化简为(sinα)(cosα)(sinα)(sinα)(cosα)sinαsinαcosα                                     答案:-\tan \alpha;提示:化简为\frac{(-\sin \alpha)(-\cos \alpha)(- \sin\alpha) (-\sin \alpha)}{(-\cos \alpha)\sin \alpha\sin \alpha\cos \alpha}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

8.求值:(1)1sin103sin80;(2)sin40(tan103);(3)2cos58+sin28cos28;             8.求值:(1)\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\sin 80^{\circ}};(2)\sin 40^{\circ}\left(\tan 10^{\circ}-\sqrt{3}\right); (3)\frac{2 \cos 58^{\circ}+\sin 28^{\circ}}{\cos 28^{\circ}};~~~~~~~~~~~~~\\
(4)sin23512cos10cos80;(5)21+sin4+2+2cos4.                                                              (4) \frac{\sin ^{2} 35^{\circ}-\displaystyle\frac{1}{2}}{\cos 10^{\circ} \cdot \cos 80^{\circ}}; (5)2 \sqrt{1+\sin 4}+\sqrt{2+2 \cos 4}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案(1)4;提示:原式=cos103sin10sin10cos10=4sin(3010)sin20=4;                              答案 (1)4; 提示:原式 =\frac{\cos 10^{\circ}-\sqrt{3} \sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}= \frac{4 \sin \left(30^{\circ}-10^{\circ}\right)}{\sin 20^{\circ}}=4 ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(2)1;提示:sin40(sin10cos103)=sin40sin103cos10cos10                         =sin402sin50cos10=2sin40cos40cos10=sin80cos10=1;(2)-1; 提示:\sin 40^{\circ} \cdot\left(\frac{\sin 10^{\circ}}{\cos 10^{\circ}}-\sqrt{3}\right)=\sin 40^{\circ} \cdot \frac{\sin 10^{\circ}-\sqrt{3} \cos 10^{\circ}}{\cos 10^{\circ}} ~~~~~~~~~~~~~~~~~~~~~~~~~\\ =\sin 40^{\circ} \cdot \frac{-2 \sin 50^{\circ}}{\cos 10^{\circ}}=\frac{-2 \sin 40^{\circ} \cdot \cos 40^{\circ}}{\cos 10^{\circ}}=\frac{-\sin 80^{\circ}}{\cos 10^{\circ}}=-1 ;
(3)3;提示:原式=2cos(30+28)+sin28cos28=3cos28cos28=3;                               (3)\sqrt{3}; 提示:原式 =\frac{2 \cos \left(30^{\circ}+28^{\circ}\right)+\sin 28^{\circ}}{\cos 28^{\circ}}=\frac{\sqrt{3} \cos 28^{\circ}}{\cos 28^{\circ}}=\sqrt{3} ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(4)1;原式=1cos70212cos10sin10=cos702sin10cos10=sin20sin20=1.                        (4)-1;原式=\frac{\displaystyle\frac{1-\cos 70^{\circ}}{2}-\frac{1}{2}}{\cos 10^{\circ} \cdot \sin 10^{\circ}} =-\frac{\cos 70^{\circ}}{2 \sin 10^{\circ} \cdot \cos 10^{\circ}}=-\frac{\sin 20^{\circ}}{\sin 20^{\circ}}=-1 .~~~~~~~~~~~~~~~~~~~~~~~~
(5)2sin2;原式=2(sin2+cos2)2+4cos22=2sin2+cos2+2cos2                 =2(sin2+cos2)2cos2=2sin2.                                 (5)2 \sin 2;原式=2 \sqrt{(\sin 2+\cos 2)^{2}}+\sqrt{4 \cos ^{2} 2} =2|\sin 2+\cos 2|+2|\cos 2|~~~~~~~~~~~~~~~~~\\=2(\sin 2+\cos 2)-2 \cos 2=2 \sin 2.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

9.(1)α+β=3π4,(1+tanα)(1+tanβ);                                                                   (2)(1+tan13)(1+tan17)(1+tan28)(1+tan32).                                           9.(1) 若 \alpha+\beta=-\frac{3 \pi}{4} , 求 (1+\tan \alpha)(1+\tan \beta);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (2)求 \left(1+\tan 13^{\circ}\right)\left(1+\tan 17^{\circ}\right)\left(1+\tan 28^{\circ}\right) \cdot\left(1+\tan 32^{\circ}\right).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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(1)答案:2;提示:tan(α+β)=tanα+tanβ1tanαtanβ=1,1tanαtanβ=tanα+tanβ,(1+tanα)(1+tanβ)=1+tanα+tanβ+tanαtanβ=2.                                     (2)答案:4;提示:(1),(1+tan13)(1+tan32)=(1+tan17)(1+tan28)=2.  (1)答案:2;提示: \boldsymbol{\operatorname { t a n }}(\alpha+\beta)=\frac{\boldsymbol{\operatorname { t a n }} \alpha+\tan \beta}{1-\boldsymbol{\operatorname { t a n }} \alpha \boldsymbol{\operatorname { t a n }} \beta}=1 , \therefore 1-\tan \alpha \boldsymbol{\operatorname { t a n }} \beta=\tan \alpha+\tan \beta , \\则 (1+\tan \alpha) \cdot(1+\tan \beta)=1+\tan \alpha+\tan \beta+\tan \alpha \boldsymbol{\operatorname { t a n }} \beta=2 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\(2)答案:4;提示:同(1),\left(1+\tan 13^{\circ}\right)\left(1+\tan 32^{\circ}\right)=\left(1+\tan 17^{\circ}\right)\left(1+\tan 28^{\circ}\right)=2 .~~

10.将下列函数化简为y=Asin(ωx+φ)+b的形式.                                                            10.将下列函数化简为y=A\sin(\omega x+\varphi )+b的形式.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(1)f(x)=2sin2ωx+23sinωxcosωx1;                                                                        (2)f(x)=(13)cos2x+sinxcosx+sin(x+π4)sin(xπ4);                                 (3)f(x)=sin(2ωxπ6)2cos2ωx+1;                                                                            (4)f(x)=cos2(x+π6)+32sin2x;                                                                                    (5)f(x)=sinωxsin(ωx+π3)sin2ωx;                                                                             (6)f(x)=cos4x2sinxcosxsin4x;                                                                              (7)f(x)=sin2(π2+x)+3sin(πx)cosxcos2x.                                                      (1)f(x)=2\sin^2\omega x+2\sqrt{3}\sin\omega x\cos\omega x-1;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (2)f(x)=(1-\sqrt{3}) \cos ^{2} x+\sin x \cos x+\sin \left(x+\frac{\pi}{4}\right) \sin \left(x-\frac{\pi}{4}\right) ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (3) f(x)=\sin \left(2 \omega x-\frac{\pi}{6}\right)-2 \cos ^{2} \omega x+1 ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (4) f(x)=\cos ^{2}\left(x+\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} \sin 2 x ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (5)f(x)=\sin \omega x \sin \left(\omega x+\frac{\pi}{3}\right)-\sin ^{2} \omega x ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (6)f(x)=\cos ^{4} x-2 \sin x \cdot \cos x-\sin ^{4} x;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (7)f(x)=\sin ^{2}\left(\frac{\pi}{2}+x\right)+\sqrt{3} \sin (\pi-x) \cos x-\cos 2 x .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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答案:(1)f(x)=2sin(2ωxπ6);                                                                                         (2)f(x)=sin(2xπ3)+132;                                                                                        (3)f(x)=3sin(2ωxπ3);                                                                                                (4)f(x)=12sin(2x+π6)+12;                                                                                              (5)f(x)=12sin(2ωx+π6)14;                                                                                            (6)f(x)=cos2xsin2x=2sin(2xπ4);                                                                  (7)f(x)=sin(2xπ6)+12.                                                                                                  答案:(1) f(x)=2 \sin \left(2\omega x-\frac{\pi}{6}\right);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (2)f(x)=\sin \left(2 x-\frac{\pi}{3}\right)+\frac{1-\sqrt{3}}{2};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (3) f(x)=\sqrt{3} \sin \left(2 \omega x-\frac{\pi}{3}\right) ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (4)f(x)=\frac{1}{2} \sin \left(2 x+\frac{\pi}{6}\right)+\frac{1}{2} ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (5)f(x)=\frac{1}{2} \sin \left(2 \omega x+\frac{\pi}{6}\right)-\frac{1}{4};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (6)f(x)=\cos 2 x-\sin 2 x=-\sqrt{2} \sin \left(2 x-\frac{\pi}{4}\right);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (7)f(x)=\sin \left(2 x-\frac{\pi}{6}\right)+\frac{1}{2}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(1)提示:降幂公式sin2ωx=1cos2ωx2f(x)=cos2ωx+3sin2ωx                 =2sin(2ωxπ6);                                                                                (2)提示:f(x)=(13)1+cos2x2+12sin2x+12(sin2xcos2x)                              =1323cos2x2+12sin2x=sin(2xπ3)+132;        (3)提示:f(x)=32sin2ωx12cos2ωxcos2ωx=3(12sin2ωx32cos2ωx) =3sin(2ωxπ3);                                                                            (4)提示:f(x)=cos2(x+π6)+32sin2x=1+cos(2x+π3)2+32sin2x                  =12+14cos2x+34sin2x=12sin(2x+π6)+12;                       (5)提示:f(x)=sinωx(sinωxcosπ3+cosωxsinπ3)sin2ωx=32cosωxsinωx 12sin2ωx=34sin2ωx14(1cos2ωx)=12sin(2ωx+π6)14;(6)提示:f(x)=cos4xsin4x2sinxcosx=(cos2xsin2x)(cos2x+sin2x)  sin2x=cos2xsin2x=2sin(2xπ4);                                  (7)提示:f(x)=1+cos2x2+32sin2xcos2x=32sin2x12cos2x+12               =sin(2xπ6)+12.                                                                            (1)提示: 降幂公式 \sin ^{2} \omega x=\frac{1-\cos 2 \omega x}{2} \Rightarrow f(x)=-\cos 2 \omega x+ \sqrt{3} \sin 2 \omega x~~~~~~~~~~~~~~~~~\\=2 \sin \left(2 \omega x-\frac{\pi}{6}\right) ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (2)提示:f(x) =(1-\sqrt{3}) \frac{1+\cos 2 x}{2}+\frac{1}{2} \sin 2 x+\frac{1}{2}\left(\sin ^{2} x-\cos ^{2} x\right) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ =\frac{1-\sqrt{3}}{2}-\frac{\sqrt{3} \cos 2 x}{2}+\frac{1}{2} \sin 2 x=\sin \left(2 x-\frac{\pi}{3}\right)+\frac{1-\sqrt{3}}{2};~~~~~~~~\\ (3) 提示:f(x) =\frac{\sqrt{3}}{2} \sin 2 \omega x-\frac{1}{2} \cos 2 \omega x-\cos 2 \omega x =\sqrt{3}\left(\frac{1}{2} \sin 2 \omega x-\frac{\sqrt{3}}{2} \cos 2 \omega x\right)~\\=\sqrt{3} \sin \left(2 \omega x-\frac{\pi}{3}\right);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \\ (4)提示: f(x)=\cos ^{2}\left(x+\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} \sin 2 x=\frac{1+\cos \left(2 x+\frac{\pi}{3}\right)}{2}+\frac{\sqrt{3}}{2} \sin 2 x ~~~~~~~~~~~~~~~~~~\\ =\frac{1}{2}+\frac{1}{4} \cos 2 x+\frac{\sqrt{3}}{4} \sin 2 x=\frac{1}{2} \sin \left(2 x+\frac{\pi}{6}\right)+\frac{1}{2} ;~~~~~~~~~~~~~~~~~~~~~~~\\ (5)提示: f(x)=\sin \omega x\left(\sin \omega x \cos \frac{\pi}{3}+\cos \omega x \sin \frac{\pi}{3}\right)-\sin ^{2} \omega x=\frac{\sqrt{3}}{2} \cos \omega x \sin \omega x-~\\\frac{1}{2} \sin ^{2} \omega x =\frac{\sqrt{3}}{4} \sin 2 \omega x-\frac{1}{4}(1-\cos 2 \omega x)=\frac{1}{2} \sin \left(2 \omega x+\frac{\pi}{6}\right)-\frac{1}{4}; \\ (6)提示: f(x)=\cos ^{4} x-\sin ^{4} x-2 \sin x \cdot \cos x=\left(\cos ^{2} x-\sin ^{2} x\right)\left(\cos ^{2} x+\sin ^{2} x\right)-~~\\\sin 2 x =\cos 2 x-\sin 2 x=-\sqrt{2} \sin \left(2 x-\frac{\pi}{4}\right);~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (7)提示: f(x)=\frac{1+\cos 2 x}{2}+\frac{\sqrt{3}}{2} \sin 2 x-\cos 2 x=\frac{\sqrt{3}}{2} \sin 2 x-\frac{1}{2} \cos 2 x+\frac{1}{2} ~~~~~~~~~~~~~~~\\ =\sin \left(2 x-\frac{\pi}{6}\right)+\frac{1}{2}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

用已知角表示所求角

1.(1)已知α为锐角,cos(α+π6)=513,cosα的值.                                                       (2)sin(θ+π6)=13,θ(0,π),求cosθ的值.                                                                    (3)ABC,内角A,B,C所对的边分别为a,b,c,sin2B=15bcosB,且角B为锐角,    b=8,sinA=13,sin(2B+C)的值.                                                                                   1.(1)已知 \alpha 为锐角, 且 \cos \left(\alpha+\frac{\pi}{6}\right)=\frac{5}{13} , 求 \cos \alpha 的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\(2)若\sin (\theta +\frac{\pi}{6})=\frac{1}{3},\theta\in(0,\pi),求\cos\theta的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (3)在 \triangle A B C 中,内角 A, B, C 所对的边分别为 a, b, c, \sin 2 B=\frac{1}{5} b \cos B ,且角 B 为锐角,~~~~\\ b=8, \sin A=\frac{1}{3} ,求 \sin (2 B+C) 的值.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:(1)53+1226;提示:cosα=cos[(α+π6)π6],其中,sin(α+π6)=1213.   答案:(1)\frac{5 \sqrt{3}+12}{26} ;提示: \cos \alpha=\cos \left[\left(\alpha+\frac{\pi}{6}\right)-\frac{\pi}{6}\right],其中,\sin \left(\alpha+\frac{\pi}{6}\right)=\frac{12}{13} .~~~
(2)答案:1266;提示:cosθ=cos[(θ+π6)π6],其中,θ(0,π),θ+π6(π6,7π6)sinπ6=12>sin(θ+π6)=13,θ+π6不能是锐角,cos(θ+π6)=223.          (2)答案:\frac{1-2\sqrt{6}}{6};提示: \cos \theta=\cos \left[\left(\theta+\frac{\pi}{6}\right)-\frac{\pi}{6}\right],其中,\because \theta\in(0,\pi),\therefore \theta+\frac{\pi}{6}\in\\ (\frac{\pi}{6},\frac{7\pi}{6})\sin\frac{\pi}{6}=\frac{1}{2}>\sin (\theta +\frac{\pi}{6})=\frac{1}{3},\therefore \theta +\frac{\pi}{6}不能是锐角,\cos(\theta +\frac{\pi}{6})=-\frac{2\sqrt{2}}{3}.~~~~~~~~~~
(3)答案:38215;提示:sinB=45,cosB=35,sinB>sinA,B>A,cosA=223.sin(2B+C)=sin(π+BA)=sin(BA).                                                                (3)答案:\frac{3-8\sqrt{2}}{15};提示:\sin B =\frac{4}{5},\cos B=\frac{3}{5},\sin B >\sin A,\therefore B > A,\cos A=\frac{2\sqrt{2}}{3}.\\ \sin (2 B+C) =\sin (\pi+B-A)=-\sin (B-A).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

2.已知sinα=255,sin(βα)=1010,α,β均为锐角,β等于(  )                                A.5π12  B.π3  Cπ4  D.π6                                                                                                               2.已知 \sin \alpha=\frac{2 \sqrt{5}}{5}, \sin (\beta-\alpha)=-\frac{\sqrt{10}}{10}, \alpha, \beta 均为锐角, 则 \beta 等于 (~~ )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ A. \frac{5 \pi}{12} ~~ B. \frac{\pi}{3} ~~ \mathrm{C} \frac{\pi}{4} ~~ D. \frac{\pi}{6} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:C;提示:sinβ=sin[α+(βα)]=sinαcos(βα)+cosαsin(βα),α,β  均为锐角,所以cosα=55,cos(βα)=31010.                                                                答案:C;提示: \sin \beta=\sin [\alpha+(\beta-\alpha)]=\sin \alpha \cdot \cos (\beta-\alpha)+\cos \alpha \cdot \sin (\beta-\alpha), \alpha, \beta ~~ \\均为锐角, 所以 \cos \alpha=\frac{\sqrt{5}}{5}, \cos (\beta-\alpha)=\frac{3 \sqrt{10}}{10} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

3.已知α,β(3π4,π),sin(α+β)=35,sin(βπ4)=2425,cos(α+π4).               3.已知 \alpha, \beta \in\left(\frac{3 \pi}{4}, \pi\right), \sin (\alpha+\beta)=-\frac{3}{5}, \sin \left(\beta-\frac{\pi}{4}\right)=\frac{24}{25} , 求 \cos \left(\alpha+\frac{\pi}{4}\right).~~~~~~~~~~~~~~~
答案

答案:45;提示:cos(α+π4)=cos[(α+β)(βπ4)],其中,α+β              (3π2,2π),cos(α+β)=45;βπ4(π2,3π4),cos(βπ4)=725.                 答案:-\frac{4}{5} ;提示: \cos \left(\alpha+\frac{\pi}{4}\right)=\cos \left[(\alpha+\beta)-\left(\beta-\frac{\pi}{4}\right)\right],其中,\because \alpha+\beta \in~~~~~~~~~~~~~~\\ \left(\frac{3 \pi}{2}, 2 \pi\right) , \therefore \cos (\alpha+\beta)=\frac{4}{5}; \because \beta-\frac{\pi}{4} \in\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) ,\therefore \cos \left(\beta-\frac{\pi}{4}\right)=-\frac{7}{25} .~~~~~~~~~~~~~~~~~

4.已知cos(θ+π4)=1010,θ(0,π2),sin(2θπ3).                                                  4. 已知 \cos \left(\theta+\frac{\pi}{4}\right)=\frac{\sqrt{10}}{10}, \theta \in\left(0, \frac{\pi}{2}\right) , 求 \sin \left(2 \theta-\frac{\pi}{3}\right).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:43310;提示:t=θ+π4,cost=1010t(π4,3π4),θ=tπ4,            sin(2θπ3)=sin(2t2π3),其中sin2t=35,cos2t=45.                                      答案:\frac{4-3 \sqrt{3}}{10} ;提示:令t=\theta+\frac{\pi}{4},则\cos t=\frac{\sqrt{10}}{10}且t \in \left ( \frac{\pi}{4}, \frac{3\pi}{4} \right ) ,\theta =t- \frac{\pi}{4}, ~~~~~~~~~~~~\\则\sin \left(2 \theta-\frac{\pi}{3}\right)= \sin \left(2 t-\frac{2\pi}{3}\right),其中\sin 2t=\frac{3}{5},\cos 2t=-\frac{4}{5}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

变式1.已知α为锐角,cos(α+π6)=45,sin(2α+π12).                                                  变式1. 已知 \alpha 为锐角, \cos \left(\alpha+\frac{\pi}{6}\right)=\frac{4}{5} , 求 \sin \left(2 \alpha+\frac{\pi}{12}\right).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:7250;提示:令α+π6=t,cost=45,sint=35,sin(2α+π12)=sin(2tπ4)  =22(sin2tcos2t)=7250.                                                                                                答案:\frac{7\sqrt{2} }{50} ;提示:令\alpha +\frac{\pi }{6} =t,则\cos t=\frac{4}{5} ,\sin t=\frac{3}{5}, \sin \left(2 \alpha+\frac{\pi}{12}\right)=\sin (2t-\frac{\pi }{4} ) ~~\\=\frac{\sqrt{2} }{2} (\sin 2t-\cos 2t)=\frac{7\sqrt{2} }{50}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
变式2.cos(π32x)=78,sin(x+π3)的值为()                                                   A.14B.78C.±14D.±78                                                                                                 变式2.若 \cos \left(\frac{\pi}{3}-2 x\right)=-\frac{7}{8} , 则 \sin \left(x+\frac{\pi}{3}\right) 的值为(\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ A. \frac{1}{4} \quad B. \frac{7}{8} \quad C. \pm \frac{1}{4} \quad D.\pm \frac{7}{8} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:C;π32x=t,cost=78,sin(x+π3)=cost2=±14.                                      答案:C;令\frac{\pi}{3}-2 x=t,\cos t=-\frac{7}{8}, \sin \left(x+\frac{\pi}{3}\right) =\cos \frac{t}{2}=\pm \frac{1}{4}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
5.已知sin(π3x)=13,0<x<π6,sin(π6+x)cos(2π3+x)的值.                 5.已知 \sin \left(\frac{\pi}{3}-x\right)=\frac{1}{3} , 0< x< \frac{\pi}{6} , 求 \sin \left(\frac{\pi}{6}+x\right)-\cos \left(\frac{2 \pi}{3}+x\right) 的值.~~~~~~~~~~~~~~~~~
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答案:423;提示:令π3x=t,π6<t<π3,x=π3tsint=13,方法同上.    答案: \frac{4 \sqrt{2}}{3} ; 提示:令 \frac{\pi}{3}-x=t, \therefore \frac{\pi}{6} < t<\frac{\pi}{3}, 且 x=\frac{\pi}{3}-t\therefore \sin t=\frac{1}{3} ,方法同上.~~~~

6.(1)求函数f(x)=3sinx+3cosx,x[0,π]的值域;                                                    (2)求函数f(x)=3sinx+4cosx,x[0,π]的值域.                                                    6.(1)求函数f(x)=3\sin x+\sqrt{3}\cos x,x\in[0,\pi]的值域;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ (2)求函数f(x)=3\sin x + 4\cos x,x\in[0,\pi]的值域.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:(1)[3,23];(2)[4,5]                                                                                           提示:(1)f(x)=23sin(x+π6),x[0,π],x+π6[π6,7π6],f(x)值域为[3,23].(2)f(x)=5(35sinx+45cosx),设cosφ=35sinφ=45φ(0,π2),                        f(x)=5sin(x+φ)x+φ[φ,π+φ]f(x)max=5,                                           f(x)min=5sin(π+φ)=5sinφ=4,f(x)值域[4,5]                                           答案: (1)[-\sqrt3,2\sqrt3];(2)[-4,5]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ 提示:(1)f(x)=2\sqrt{3}\sin(x + \frac{\pi}{6}),x\in[0,\pi],x+\frac{\pi}{6}\in[\frac{\pi}{6},\frac{7\pi}{6}], f(x)值域为[-\sqrt3,2\sqrt3].\\ (2)f(x)=5(\frac{3}{5}\sin x+\frac{4}{5}\cos x),设\cos\varphi=\frac{3}{5},\sin\varphi=\frac{4}{5},\varphi\in(0,\frac{\pi}{2}),~~~~~~~~~~~~~~~~~~~~~~~~\\ \therefore f(x)=5\sin(x + \varphi),x+\varphi\in[\varphi,\pi+\varphi],f(x)_{\max}=5,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ f(x)_{\min}=5\sin(\pi+\varphi)= - 5\sin\varphi=-4, \therefore f(x)值域[-4,5]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

7.若函数f(x)=2sinx+cosx3,x(0,π)的两个零点分别为x1x2,                cos(x1x2)                                                                                                                     7.若函数 f(x)=2 \sin x+\cos x-\sqrt{3}, x \in(0, \pi) 的两个零点分别为 x_{1} 和 x_{2} ,~~~~~~~~~~~~~~~~\\求 \cos \left(x_{1}-x_{2}\right) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
答案

答案:(1)[3,23];(2)[4,5]                                                                                           提示:(1)f(x)=23sin(x+π6),x[0,π],x+π6[π6,7π6],f(x)值域为[3,23].(2)f(x)=5(35sinx+45cosx),设cosφ=35sinφ=45φ(0,π2),                        f(x)=5sin(x+φ)x+φ[φ,π+φ]f(x)max=5,                                           f(x)min=5sin(π+φ)=5sinφ=4,f(x)值域[4,5]                                           答案: (1)[-\sqrt3,2\sqrt3];(2)[-4,5]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ 提示:(1)f(x)=2\sqrt{3}\sin(x + \frac{\pi}{6}),x\in[0,\pi],x+\frac{\pi}{6}\in[\frac{\pi}{6},\frac{7\pi}{6}], f(x)值域为[-\sqrt3,2\sqrt3].\\ (2)f(x)=5(\frac{3}{5}\sin x+\frac{4}{5}\cos x),设\cos\varphi=\frac{3}{5},\sin\varphi=\frac{4}{5},\varphi\in(0,\frac{\pi}{2}),~~~~~~~~~~~~~~~~~~~~~~~~\\ \therefore f(x)=5\sin(x + \varphi),x+\varphi\in[\varphi,\pi+\varphi],f(x)_{\max}=5,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ f(x)_{\min}=5\sin(\pi+\varphi)= - 5\sin\varphi=-4, \therefore f(x)值域[-4,5]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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