求sinα－cosα>0 sinα-cosα＜0的证明过程
\u5df2\u77e5sin\u03b1cos\u03b1<0\uff0c\u5219\u03b1\u662f\u7b2c\u51e0\u8c61\u9650\u89d2\uff0c\u6c42\u8fc7\u7a0b\uff0c\u8c22\u8c22\u89e3\u7b54\uff1a
\u5df2\u77e5sin\u03b1cos\u03b1<0
\u2234sin\u03b1\u548ccos\u03b1\u5f02\u53f7
\u2234 \u03b1\u662f\u7b2c\u4e8c\u6216\u7b2c\u56db\u8c61\u9650\u89d2
sin\u03b1cos\u03b1<0
\u5f97sina\uff1e0\uff0ccosa\uff1c0
\u6216sina\uff1c0\uff0ccosa\uff1e0
sina\u5728\u4e00\u3001\u4e8c\u8c61\u9650\u4e3a\u6b63\uff0c\u4e09\u3001\u56db\u8c61\u9650\u4e3a\u8d1f
cosa\u5728\u4e00\u3001\u56db\u8c61\u9650\u4e3a\u6b63\uff0c\u4e8c\u3001\u4e09\u8c61\u9650\u4e3a\u8d1f
\u6240\u4ee5
\u5f53sina\uff1e0\uff0ccosa\uff1c0\u65f6\uff0ca\u5728\u7b2c\u4e8c\u8c61\u9650
\u5f53sina\uff1c0\uff0ccosa\uff1e0\u65f6\uff0ca\u5728\u7b2c\u56db\u8c61\u9650
\u7efc\u4e0a\u53ef\u5f97\uff1a\u03b1\u7684\u7ec8\u8fb9\u843d\u5728\u7b2c\u4e8c\u8c61\u9650\u6216\u7b2c\u56db\u8c61\u9650
2k∏+∏/4<a<2k∏+5∏/4,sinα－cosα>0
2k∏-3∏/4<a<2k∏+∏/4 ,sinα-cosα＜0
1)
sinα－cosα>0
√2sin(a-∏/4)>0
0<a-∏/4<∏
2k∏+∏/4<a<2k∏+5∏/4
2)
sinα－cosα<0
√2sin(a-∏/4)<0
-∏<a-∏/4<0
2k∏-3∏/4<a<2k∏+∏/4
sina-cosa
=√2(sina/√2-cosa/√2)
=√2(cosπ/4sina-cosasinπ/4)
=√2sin(a-π/4)
sina-cosa>0,即sin(a-π/4)>0
2kπ<a-π/4<2kπ+π
2kπ+π/4<a<2kπ+5π/4
sina-cosa<0,即sin(a-π/4)<0
2kπ-π<a-π/4<2kπ
2kπ-3π/4<a<2kπ+π/4
用三角函数线画图
绛旓細sin伪-cos伪=鈭2(鈭2/2*sin伪-鈭2/2*cos伪)=鈭2[cos(蟺/4)*sin伪-sin(蟺/4)*cos伪)]=鈭2sin(伪-蟺/4)
绛旓細浠=sin伪-cos伪锛屾牴鎹渶鍊煎彧鑳藉嚭鐜板湪绔偣鍜屾瀬鍊肩偣锛屽厛姹傚y'=cos伪+sin伪锛屽湪[pi/2锛宲i]涓紝褰撲笖浠呭綋伪=3pi/4鏃讹紝y'=0 绔偣pi/2, y=1-0=1锛岀鐐筽i锛寉=1-(-1)=1锛屾瀬鍊肩偣3pi/4锛寉=鈭2/2-(-鈭2/2)=鈭2 鍥犳sin伪-cos伪鐨勫彇鍊艰寖鍥碵1,鈭2]pi鈥︹﹀渾鍛ㄧ巼 ...
绛旓細(sin伪-cos伪)²=2 sin²伪+cos²伪-2sin伪cos伪=2 sin伪cos伪=-1/2 (1) tan伪+cot伪 =sin伪/cos伪+cos伪/sin伪 =(sin²伪+cos²伪)/(sin伪cos伪)=1/(-1/2)=-2 (2)sin³伪-cos³伪 =(sin伪-cos伪)(sin²伪+cos²伪+...
绛旓細sin伪-cos伪=鈭2 骞虫柟 (sin伪-cos伪)²=2 sin²伪-2sin伪cos伪+cos²伪=2 1-2sin伪cos伪=2 2sin伪cos伪=-1 sin伪cos伪=-1/2 sin³伪-cos³伪 =(sin伪-cos伪)(sin²伪+sin伪cos伪+cos²伪)=鈭2(1+sin伪cos伪)=鈭2(1-1/2)=鈭2/2...
绛旓細鎻愬嚭涓牴鍙2,鏍瑰彿2*sin(伪-pi/4)鎵浠 澶т簬绛変簬-鏍瑰彿2 灏忎簬绛変簬鏍瑰彿2
绛旓細sin伪-cos伪=1/2 骞虫柟 sin²伪-2sin伪cos伪+cos²伪=1/4 1-2sin伪cos伪=1/4 2sin伪cos伪=3/4 sin伪cos伪=3/8 sin³伪-cos³伪 =(sin伪-cos伪)(sin²伪+sin伪cos伪+cos²伪)=1/2*(1+3/8)=1/2*11/8 =11/16 ...
绛旓細sin(伪+伪)=sina伪cos伪+cos伪sin伪=sina2伪 sin伪-cos伪=1/3 鎵浠ワ紙sin伪-cos伪)^2=1/9 sin伪^2+cos伪^2-2sin伪cos伪=1/9 1-2sin伪cos伪=1/9 2sin伪cos伪=8/9 鎵浠 sin2伪= 2sin伪*cos伪=8/9 濡傛灉甯埌浣狅紝璇疯寰楅噰绾筹紝O(鈭鈭)O璋㈣阿 ...
绛旓細鍏堟妸鍓嶉潰鐨勫紡瀛愰兘骞虫柟 姹傚嚭sin闃垮皵娉*cos闃垮皵娉 鍐嶅皢鍚庨潰鐨勫紡瀛愪篃骞虫柟 姹傚嚭sin闃垮皵娉-cos闃垮皵娉曠殑骞虫柟绛変簬 sin闃垮皵娉-cos闃垮皵娉曠瓑浜庢牴鍙蜂笅2-M鐨勫钩鏂 鍦ㄦ牴鎹綘鐨勯偅涓鐩樋灏旀硶鐨勭鍙峰湪鐪嬫槸姝ｆ槸璐
绛旓細sin伪+cos伪=0鎺ㄥ嚭sin伪=-cos伪 sin伪^2+cos伪^2=1 灏嗙涓涓紡瀛愬甫鍏ュ埌绗簩涓紡瀛愪腑 鏈夊綋sin伪=鏍瑰彿2/2鏃 cos伪=-鏍瑰彿2/2 褰搒in伪=-鏍瑰彿2/2鏃 cos伪=鏍瑰彿2/2
绛旓細sin伪-cos伪锛濇牴鍙2锛1-2sin伪cos伪=2 sin伪cos伪=1/2 tan伪锛1锛弔an伪 =sin伪/cos伪+cos伪/sin伪 =1/cos伪sin伪 =1/(1/2)=2
