Question

高中数学必修4，诱导公式问题

1. sin^2(1°)+sin^2(2°)+sin^2(3°)+……+sin^2(89°)+sin^2(90°).
2.已知tanα100°=k,则sin80°=_______?
3.已知α是锐角,2tan(π-α)-3cos(π/2+β)+5=0,tan(π+α)+6sin(π+β)-1=0.则sinα=______?
详细步骤

Answer 1

解：1.设A=sin^2(1°)+sin^2(2°)+sin^2(3°)+……+sin^2(89°)+sin^2(90°).① 则

A=sin^2(90°)+sin^2(89°)+sin^2(88°)+……+sin^2(2°)+sin^2(1°)
=sin^2(90°)+cos^2(1°)+cos^2(2°)+……+cos^2(88°)+cos^2(89°).②

把①②错位相加，即
2A=sin^2(90°）+[sin^2(1°)+cos^2(1°)]+[sin^2(2°)+cos^2(2°)]
+[sin^2(3°)+cos^2(3°)]+……+[sin^2(89°)+cos^2(89°)]+sin^2(90°)
=1+89*1+1=91
A=91/2

2.由sin(180°-A)=sinA,cos(180°-A)=-cosA，得tan(180°-A)=-tanA.则
tan80°=tan(180°-100°)=-tan100°=-k.
sin^2(80°)/[1-sin^2(80°)]=k^2,解得sin80°=k/(1+k^2的算术平方根)

3.由cos(90°+A)=-sinA,sin(180°+A)=-sinA,tan(180°-A)=-tanA tan(180°+A)=tanA .得
2tan(π-α)-3cos(π/2+β)+5=0 即 -2tanα+3sinβ+5=0 ①
tan(π+α)+6sin(π+β)-1=0 即 tanα-6sinβ-1=0 ②
①*2+②得tanπα=3，则 sinα=3/(10的算术平方根)

Answer 2

<img src="http://i159.photobucket.com/albums/t145/l421013/math/s5r.jpg">
<img src="http://i159.photobucket.com/albums/t145/l421013/math/t10.jpg">
<img src="http://i159.photobucket.com/albums/t145/l421013/math/0q1.jpg">

Answer 3

1.sin^2(1)+sin^2(89)=1,sin^2(2)+sin^(88)=16……，sin^2(44)+sin^2(46)=1
共44,sin^2(45)=1/2,sin^2(90)=1,总共45。5 （“度”不会打,抱歉。下同）
2。sin80=sin100,sin80=tan100/根号（1+tan^2(100)=k/根号(1+k^2)