Question

数学三角函数恒等变化怎么用只含有半角的正切的式子分别表示角的正弦、余弦和正切,

Answer 1

sinx=2sin(x/2)cos(x/2)=2tan(x/2)cos^2(x /2);

cosx=cos^2(x /2)-sin^2 (x /2)=(1-tan^2(x /2))cos^2 (x /2);

tanx=2tan(x /2)/(1-tan^2 (x /2));

cos^2(x /2)+sin^2 (x /2)=1;

1+tan^2 (x /2))=1/cos^2 (x /2);

故cos^2(x /2)=1/(tan^2(x /2)+1),代入

得sinx=2tan(x/2)cos^2(x /2)=2tan(x/2)/(tan^2(x /2)+1)

cosx=(1-tan^2(x /2))cos^2 (x /2)=(1-tan^2(x /2))/(tan^2(x /2)+1)

cotx=1/tanx=(1-tan^2 (x /2))/(2tan(x /2))