已知sin(π-a)-cos(π a)=根号2/3,(π/2小于a小于π)求sina-cosa的值
根据已知条件,sin(π-a)-cos(π a)=sqrt(2)/3, 因为sin(π-a)=sin(a), cos(π a)=-cos(a), 所以上式等价于 sin(a) cos(a)=sqrt(2)/3,(*) 因为sin^2(a) cos^2(a)=1, 所以 (sin(a)-cos(a))^2 =2*(sin^2(a) cos^2(a))-(sin(a) cos(a))^2 =2*1-(sqrt(2)/3)^2 =2-2/9 =16/9, 并且由于π/2小于a小于π,故sin(a) > 0 > cos(a), 所以sin(a)-cos(a)>0,故 sin(a)-cos(a)=...全部
根据已知条件,sin(π-a)-cos(π a)=sqrt(2)/3, 因为sin(π-a)=sin(a), cos(π a)=-cos(a), 所以上式等价于 sin(a) cos(a)=sqrt(2)/3,(*) 因为sin^2(a) cos^2(a)=1, 所以 (sin(a)-cos(a))^2 =2*(sin^2(a) cos^2(a))-(sin(a) cos(a))^2 =2*1-(sqrt(2)/3)^2 =2-2/9 =16/9, 并且由于π/2小于a小于π,故sin(a) > 0 > cos(a), 所以sin(a)-cos(a)>0,故 sin(a)-cos(a)=sqrt(16/9)=4/3。
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