三角函數/5倍角公式

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公式[编辑]

sinα=y,cosα=x

sin5α=16y5-20y3+5y
cos5α=16x5-20x3+5x

證明[编辑]

cos5α=cos(3α+2α)=cos3α×cos2α-sin3α×sin2α=(4x3-3x)(2x2-1)-(3y-4y3)(2xy)

=8x5-4x3-6x3+3x-2x(3y2-4y4)=8x5-10x3+3x-2x[3(1-x2)-4(1-x2)2]
=8x5-10x3+3x-2x[3-3x2-4(1-2x2+x4)]=8x5-10x3+3x-2x[-1+5x2-4x4]
=8x5-10x3+3x+[2-10x3+8x5]=16x5-20x3+5x


sin5α=sin(3α+2α)=cos3α×sin2α+sin3α×cos2α=(4x3-3x)(2xy)+(3y-4y3)(1-2y2)

=(4x2-3)(2x2y)+8y5-4y3-6y3+3y=[4(1-y2)-3]×2(1-y2)y+8y5-10y3+3y
=[4-4y2-3](2y-2y3)+8y5-10y3+3y=[1-4y2](2y-2y3)+8y5-10y3+3y
=8y5-10y3+2y+8y5-10y3+3y=16y5-20y3+5y

求值[编辑]

令 sinα=y,cosα=x

(一)五倍角之正弦、餘弦值等於一倍角[编辑]

x=16x5-20x3+5x

0=16x5-20x3+4x=4x(4x4-5x2+1)

=4x(4x2-1)(x2-1)
=4x(2x+1)(2x-1)(x+1)(x-1)

x=0,±½,±1,∴±90°,±60°,0°的cos5α=cosα

同理,
y=0,±½,±1,∴0°,±30°,±90°的sin5α=sinα

(二)五倍角之正弦、餘弦值等於負一倍角[编辑]

-x=16x5-20x3+5x

0=16x5-20x3+6x=2x(8x4-10x2+3)

=2x(4x2-3)(2x2-1)
=2x(2x+√3)(2x-√3)(√2x+1)(√2x-1)

x=0,±√3/2,±√2/2,∴±90°,±30°,±45°的cos5α=-cosα

同理,
y=0,±√3/2,±√2/2,∴0°,±60°,±45°的sin5α=-sinα

(三)五倍角之正弦、餘弦值等於 1[编辑]

1=16x5-20x3+5x

0=16x5-20x3+5x-1=(x-1)(16x4+16x3-4x2-4x+1)

=(4x2+2x-1)2(x-1)

x=1,,∵0°,72°,144°的cos5α=1∴cos72°=,cos144°=
∴sin18°=cos72°=,sin54°=cos36°=-cos144°=
∴sin36°=cos54°=√1-sin254°=
∴sin72°=cos18°=√1-sin218°=

同理,
1=16y5-20y3+5y
y=1,,∵90°,18°,-54°的sin5α=1∴sin18°=,sin-54°=
∴sin18°=cos72°=,sin54°=-sin-54°=

(四)五倍角之正弦、餘弦值等於 0[编辑]

0=16x5-20x3+5x=x(16x4-20x2+5)
x2==>x=
∵90°,18°,54°,126°,162°的cos5α=0,
∴cos18°=、cos162°=、cos54°=、cos126°=

同理,
0=16y5-20y3+5y=y(16y4-20y2+5)
y2==>y=
∵0°,±36°,±72°的sin5α=0,
∴sin72°=、sin-72°=、sin36°=、sin-36°=