10.1 Let
θ = π 10 = 1 8 0 5 θ = π 2 cos 3 θ = sin 2 θ cos 5 4 0 = sin 3 6 0 = sin ( 9 0 0 − 5 4 0 ) 4 cos 3 θ − 3 cos θ = 2 sin θ cos θ 4 cos 2 θ − 3 = 2 sin θ 4 ( 1 − sin 2 θ ) − 3 = 2 sin θ 4 sin 2 θ + 2 sin θ − 1 = 0 sin θ = 5 − 1 4 cos 2 θ = cos π 5 = 1 − 2 sin 2 θ = 5 + 1 4 cos π 5 = 5 + 1 4 sin π 5 = 1 − cos 2 θ = 1 − ( 5 + 1 4 ) 2 = 10 − 2 5 4 sin 2 π 5 = 2 sin π 5 cos π 5 = 10 − 2 5 4 5 + 1 2 = = 10 + 2 5 4 \theta = \frac {\pi} {10} = 18 ^ 0\\
5 \theta = \frac {\pi} {2}\\
\cos3 \theta = \sin2 \theta \\
\cos54 ^ 0 = \sin36 ^ 0 = \sin (90 ^ 0-54 ^ 0) \\
4 \cos ^ 3 \theta-3 \cos \theta = 2 \sin \theta \cos \theta \\
4 \cos ^ 2 \theta-3 = 2 \sin \theta \\
4 (1- \sin ^ 2 \theta) -3 = 2 \sin \theta \\
4 \sin ^ 2 \theta + 2 \sin \theta-1 = 0 \\
\sin \theta = \frac {\sqrt5-1} {4} \\
\cos2 \theta = \cos \frac {\pi} {5} = 1-2 \sin ^ 2 \theta = \frac {\sqrt5 + 1} {4} \\
\cos \frac {\pi} {5}= \frac {\sqrt5 + 1} {4}\\
\sin \frac {\pi} {5} = \sqrt {1- \cos ^ 2 \theta} = \sqrt {1 - (\frac {\sqrt5 + 1} {4}) ^ 2} = \frac {\sqrt {10-2 \sqrt5}} {4} \\
\sin \frac {2 \pi} {5 } = 2 \sin \frac {\pi} {5} \cos \frac {\pi} {5} = \frac {\sqrt {10-2 \sqrt5}} {4} \frac {\sqrt5 + 1} {2} = \\
= \frac {\sqrt {10 + 2 \sqrt5}} {4} θ = 10 π = 1 8 0 5 θ = 2 π cos 3 θ = sin 2 θ cos 5 4 0 = sin 3 6 0 = sin ( 9 0 0 − 5 4 0 ) 4 cos 3 θ − 3 cos θ = 2 sin θ cos θ 4 cos 2 θ − 3 = 2 sin θ 4 ( 1 − sin 2 θ ) − 3 = 2 sin θ 4 sin 2 θ + 2 sin θ − 1 = 0 sin θ = 4 5 − 1 cos 2 θ = cos 5 π = 1 − 2 sin 2 θ = 4 5 + 1 cos 5 π = 4 5 + 1 sin 5 π = 1 − cos 2 θ = 1 − ( 4 5 + 1 ) 2 = 4 10 − 2 5 sin 5 2 π = 2 sin 5 π cos 5 π = 4 10 − 2 5 2 5 + 1 = = 4 10 + 2 5
10.2
z = cos θ + i sin θ z n = cos n θ + i sin n θ z − n = 1 cos n θ + i sin n θ = cos n θ − i sin n θ n ∈ N ( а ) z n + z − n = 2 cos n θ z n − z − n = 2 i sin n θ z = \cos \theta + i \sin \theta \\
z ^ n = \cos n \theta + i \sin n \theta \\
z ^ {- n} = \frac {1} {\cos n \theta + i \sin n \theta} = \cos n \theta-i \sin n \theta \\
n \in N\\
(а)\\
z ^ n + z ^ {- n} = 2 \cos n \theta \\
z ^ n-z ^ {- n} = 2i \sin n \theta z = cos θ + i sin θ z n = cos n θ + i sin n θ z − n = c o s n θ + i s i n n θ 1 = cos n θ − i sin n θ n ∈ N ( а ) z n + z − n = 2 cos n θ z n − z − n = 2 i sin n θ
( b ) z = cos θ + i sin θ ( z + 1 ) n = ( cos θ + i sin θ + 1 ) n = = ( 2 cos 2 θ 2 + i 2 sin θ 2 cos θ 2 ) n = = 2 n cos n θ 2 ( cos θ 2 + i sin θ 2 ) n = = 2 n cos n θ 2 ( cos n θ 2 + i sin n θ 2 ) = = 2 n cos n θ 2 z n 2 2 n cos n θ 2 z n 2 = 2 n cos n θ ( z − 1 ) n = ( cos θ + i sin θ − 1 ) n = = ( 2 sin 2 θ 2 + i 2 sin θ 2 cos θ 2 ) n = = 2 n sin n θ 2 ( sin θ 2 + i cos θ 2 ) n = = 2 n sin n θ 2 ( i ( cos θ 2 + i sin θ 2 ) ) n = = ( 2 i ) n sin n θ 2 ( cos n θ 2 + i sin n θ 2 ) = = ( 2 i ) n sin n θ 2 z n 2 ( 2 i ) n sin n θ 2 z n 2 = ( 2 i ) n sin n θ (b)\\
z = \cos \theta + i \sin \theta \\
(z + 1) ^ n = (\cos \theta + i \sin \theta + 1) ^ n = \\
= (2 \cos ^ 2 \frac {\theta} {2} + i2 \sin \frac {\theta} {2} \cos \frac {\theta} {2}) ^ n = \\
= 2 ^ n \cos ^ n \frac {\theta } {2} (\cos \frac {\theta} {2} + i \sin \frac {\theta} {2}) ^ n =
\\ = 2 ^ n \cos ^ n \frac {\theta} { 2} (\cos \frac {n \theta} {2} + i \sin \frac {n \theta} {2}) = \\
= 2 ^ n \cos ^ n \frac {\theta} {2} z ^ {\frac {n} {2}} \\
2 ^ n \cos ^ n \frac {\theta} {2} z ^ {\frac {n} {2}} = 2 ^ n \cos ^ n \theta \\
\\\\
(z-1) ^ n = (\cos \theta + i \sin \theta-1) ^ n = \\
= (2 \sin ^ 2 \frac {\theta} {2} + i2 \sin \frac { \theta} {2} \cos \frac {\theta} {2}) ^ n = \\
= 2 ^ n \sin ^ n \frac {\theta} {2} (\sin \frac {\theta} { 2} + i \cos \frac {\theta} {2}) ^ n = \\
= 2 ^ n \sin ^ n \frac {\theta} {2} ( i (\cos \frac {\theta} {2} + i \sin \frac {\theta} {2})) ^ n = \\
= (2i) ^ n \sin ^ n \frac {\theta} {2} (\cos \frac {n \theta} {2} + i \sin \frac {n \theta} {2}) = \\
= (2i) ^ n \sin ^ n \frac {\theta} {2} z ^ {\frac {n} {2}} \\
(2i) ^ n \sin ^ n \frac {\theta} {2} z ^ {\frac {n} {2}} = (2i) ^ n \sin ^ n \theta \\ ( b ) z = cos θ + i sin θ ( z + 1 ) n = ( cos θ + i sin θ + 1 ) n = = ( 2 cos 2 2 θ + i 2 sin 2 θ cos 2 θ ) n = = 2 n cos n 2 θ ( cos 2 θ + i sin 2 θ ) n = = 2 n cos n 2 θ ( cos 2 n θ + i sin 2 n θ ) = = 2 n cos n 2 θ z 2 n 2 n cos n 2 θ z 2 n = 2 n cos n θ ( z − 1 ) n = ( cos θ + i sin θ − 1 ) n = = ( 2 sin 2 2 θ + i 2 sin 2 θ cos 2 θ ) n = = 2 n sin n 2 θ ( sin 2 θ + i cos 2 θ ) n = = 2 n sin n 2 θ ( i ( cos 2 θ + i sin 2 θ ) ) n = = ( 2 i ) n sin n 2 θ ( cos 2 n θ + i sin 2 n θ ) = = ( 2 i ) n sin n 2 θ z 2 n ( 2 i ) n sin n 2 θ z 2 n = ( 2 i ) n sin n θ
( c ) sin θ = z − z − 1 2 i sin 7 θ = ( z − z − 1 ) 7 ( 2 i ) 7 = = 1 ( 2 i ) 7 ( z 7 − 7 z 6 z − 1 + 21 z 5 z − 2 − 35 z 4 z − 3 + + 35 z 3 z − 4 − 21 z 2 z − 5 + 7 z z − 6 − z − 7 ) = = 1 ( 2 i ) 7 ( 2 i sin 7 θ − 7 ⋅ ( 2 i ) sin 5 θ + + 21 ⋅ ( 2 i ) sin 3 θ − 35 ( 2 i ) sin θ ) = = 1 ( 2 i ) 6 ( sin 7 θ − 7 sin 5 θ + 21 sin 3 θ − 35 s i n θ ) = = − 1 64 ( sin 7 θ − 7 sin 5 θ + 21 sin 3 θ − 35 s i n θ ) (c)\\
\sin \theta = \frac {z-z ^ {- 1}} {2i} \\
\sin ^ 7 \theta = \frac {(z-z ^ {- 1}) ^ 7} {(2i)^7} = \\
= \frac {1} {(2i)^7} (z ^ 7-7z ^ 6z ^ {- 1} + 21z ^ 5z ^ {- 2} -35z ^ 4z ^ {- 3} + \\ + 35z ^ 3z ^ {- 4} - 21z ^ 2z ^ {- 5} + 7zz ^ {- 6} -z ^ {- 7}) = \\
= \frac {1} {(2i)^7} (2i \sin7 \theta-7 \cdot (2i) \sin5 \theta +\\
+ 21 \cdot (2i) \sin3 \theta - 35 (2i) \sin \theta) = \\
=\frac{1}{(2i)^6} (\sin7 \theta-7 \sin5 \theta + 21 \sin3 \theta-35 \ sin\theta)=\\
=-\frac{1}{64} (\sin7 \theta-7 \sin5 \theta + 21 \sin3 \theta-35 \ sin\theta)\\ ( c ) sin θ = 2 i z − z − 1 sin 7 θ = ( 2 i ) 7 ( z − z − 1 ) 7 = = ( 2 i ) 7 1 ( z 7 − 7 z 6 z − 1 + 21 z 5 z − 2 − 35 z 4 z − 3 + + 35 z 3 z − 4 − 21 z 2 z − 5 + 7 z z − 6 − z − 7 ) = = ( 2 i ) 7 1 ( 2 i sin 7 θ − 7 ⋅ ( 2 i ) sin 5 θ + + 21 ⋅ ( 2 i ) sin 3 θ − 35 ( 2 i ) sin θ ) = = ( 2 i ) 6 1 ( sin 7 θ − 7 sin 5 θ + 21 sin 3 θ − 35 s in θ ) = = − 64 1 ( sin 7 θ − 7 sin 5 θ + 21 sin 3 θ − 35 s in θ )
( d ) cos 3 θ = z 3 + z − 3 2 = ( z + z − 1 ) ( z 2 − z z − 1 + z − 2 ) 2 = = 2 cos θ ( 2 cos 2 θ − 1 ) 2 = cos θ ( 2 cos 2 θ − 1 ) sin 4 θ = z 4 − z − 4 2 i = ( z 2 + z − 2 ) ( z 2 − z − 2 ) 2 i = 2 cos 2 θ 2 i sin 2 θ 2 i = = 2 cos 2 θ sin 2 θ (d)\\
\cos3 \theta = \frac {z ^ 3 + z ^ {- 3}} {2} = \frac {(z + z ^ {- 1}) (z ^ 2-zz ^ {- 1} + z ^ {-2})} {2} = \\
= \frac {2 \cos \theta (2 \cos2 \theta-1)} {2} = \cos \theta (2 \cos2 \theta-1) \\
\sin4 \theta = \frac {z ^ 4-z ^ {- 4}} {2i} = \frac {(z ^ 2 + z ^ {- 2}) (z ^ 2-z ^ {- 2}) } {2i} = \frac {2 \cos2 \theta2i \sin2 \theta} {2i} = \\
= 2 \cos2 \theta \sin2 \theta ( d ) cos 3 θ = 2 z 3 + z − 3 = 2 ( z + z − 1 ) ( z 2 − z z − 1 + z − 2 ) = = 2 2 c o s θ ( 2 c o s 2 θ − 1 ) = cos θ ( 2 cos 2 θ − 1 ) sin 4 θ = 2 i z 4 − z − 4 = 2 i ( z 2 + z − 2 ) ( z 2 − z − 2 ) = 2 i 2 c o s 2 θ 2 i s i n 2 θ = = 2 cos 2 θ sin 2 θ
( е ) cos 3 θ = z 3 + z − 3 2 = ( z + z − 1 ) ( z 2 − z z − 1 + z − 2 ) 2 = = 2 c o s θ ( 2 cos 2 θ − 1 ) 2 = cos θ ( 4 cos 2 θ − 3 ) = = 4 cos 3 θ − 3 cos θ 4 x = cos 3 θ + 3 cos θ 4 x = 4 cos 3 θ − 3 cos θ + 3 cos θ x = cos 3 θ sin 3 θ = z 3 − z − 3 2 i = ( z z − 1 ) ( z 2 + z z − 1 + z − 2 ) 2 i = = 2 i sin θ ( 2 cos 2 θ + 1 ) 2 i = sin θ ( 3 − 4 s i n 2 θ ) = = 3 sin θ − 4 sin 3 θ 4 y = 3 sin θ − sin 3 θ 4 y = 3 sin θ − 3 sin θ + 4 sin 3 θ y = sin 3 θ (е)\\
\cos3 \theta = \frac {z ^ 3 + z ^ {- 3}} {2} = \frac {(z + z ^ {- 1}) (z ^ 2-zz ^ {- 1} + z ^ {-2})} {2} = \\
= \frac {2 \ cos\theta (2 \cos2 \theta-1)} {2} = \cos \theta (4 \cos ^ 2 \theta-3) = \\
= 4 \cos ^ 3 \theta-3 \cos \theta \\
4x = \cos3 \theta + 3 \cos \theta \\
4x = 4 \cos ^ 3 \theta-3 \cos \theta + 3 \cos \theta \\
x = \cos ^ 3 \theta\\
\sin3 \theta = \frac {z ^ 3-z ^ {- 3}} {2i} = \frac {(zz ^ {- 1}) (z ^ 2 + zz ^ {- 1} + z ^ {- 2})} {2i} = \\ =
\frac {2i \sin \theta (2 \cos2 \theta + 1)} {2i} = \sin \theta (3-4 \ sin ^ 2 \theta) = \ \
= 3 \sin \theta-4 \sin ^ 3 \theta \\
4y = 3 \sin \theta- \sin3 \theta \\
4y = 3 \sin \theta-3 \sin \theta + 4 \sin ^ 3 \theta \\
y = \sin ^ 3 \theta ( е ) cos 3 θ = 2 z 3 + z − 3 = 2 ( z + z − 1 ) ( z 2 − z z − 1 + z − 2 ) = = 2 2 cos θ ( 2 c o s 2 θ − 1 ) = cos θ ( 4 cos 2 θ − 3 ) = = 4 cos 3 θ − 3 cos θ 4 x = cos 3 θ + 3 cos θ 4 x = 4 cos 3 θ − 3 cos θ + 3 cos θ x = cos 3 θ sin 3 θ = 2 i z 3 − z − 3 = 2 i ( z z − 1 ) ( z 2 + z z − 1 + z − 2 ) = = 2 i 2 i s i n θ ( 2 c o s 2 θ + 1 ) = sin θ ( 3 − 4 s i n 2 θ ) = = 3 sin θ − 4 sin 3 θ 4 y = 3 sin θ − sin 3 θ 4 y = 3 sin θ − 3 sin θ + 4 sin 3 θ y = sin 3 θ
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