三角函數/5倍角公式

公式

sinα=y，cosα=x

sin5α=16y⁵-20y³+5y

cos5α=16x⁵-20x³+5x

證明

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

令 sinα=y，cosα=x

(一)五倍角之正弦、餘弦值等於一倍角

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

-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

1=16x⁵-20x³+5x

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

=(4x²+2x-1)²(x-1)

x=1,${\frac {-1\pm {\sqrt {5}}}{4}}$，∵0°,72°,144°的cos5α=1∴cos72°=${\frac {-1+{\sqrt {5}}}{4}}$，cos144°=${\frac {-1-{\sqrt {5}}}{4}}$
∴sin18°=cos72°=${\frac {-1+{\sqrt {5}}}{4}}$，sin54°=cos36°=-cos144°=${\frac {1+{\sqrt {5}}}{4}}$
∴sin36°=cos54°=√1-sin²54°=${\sqrt {\frac {5-{\sqrt {5}}}{8}}}$
∴sin72°=cos18°=√1-sin²18°=${\sqrt {\frac {5+{\sqrt {5}}}{8}}}$

同理，
1=16y⁵-20y³+5y
y=1,${\frac {-1\pm {\sqrt {5}}}{4}}$，∵90°,18°,-54°的sin5α=1∴sin18°=${\frac {-1+{\sqrt {5}}}{4}}$，sin-54°=${\frac {-1-{\sqrt {5}}}{4}}$
∴sin18°=cos72°=${\frac {-1+{\sqrt {5}}}{4}}$，sin54°=-sin-54°=${\frac {1+{\sqrt {5}}}{4}}$

(四)五倍角之正弦、餘弦值等於 0

0=16x⁵-20x³+5x=x(16x⁴-20x²+5)
x²=${\frac {5\pm {\sqrt {5}}}{8}}$=>x=$\pm {\sqrt {\frac {5\pm {\sqrt {5}}}{8}}}$
∵90°,18°,54°,126°,162°的cos5α=0，
∴cos18°=${\sqrt {\frac {5+{\sqrt {5}}}{8}}}$、cos162°=$-{\sqrt {\frac {5+{\sqrt {5}}}{8}}}$、cos54°=${\sqrt {\frac {5-{\sqrt {5}}}{8}}}$、cos126°=$-{\sqrt {\frac {5-{\sqrt {5}}}{8}}}$

同理，
0=16y⁵-20y³+5y=y(16y⁴-20y²+5)
y²=${\frac {5\pm {\sqrt {5}}}{8}}$=>y=$\pm {\sqrt {\frac {5\pm {\sqrt {5}}}{8}}}$
∵0°,±36°,±72°的sin5α=0，
∴sin72°=${\sqrt {\frac {5+{\sqrt {5}}}{8}}}$、sin-72°=$-{\sqrt {\frac {5+{\sqrt {5}}}{8}}}$、sin36°=${\sqrt {\frac {5-{\sqrt {5}}}{8}}}$、sin-36°=$-{\sqrt {\frac {5-{\sqrt {5}}}{8}}}$