It is known that I is a mathematical unit, if the complex number Z1 = cos α + isin α. Z2 = cos β + isin β satisfies | z1-z2 | 2 times the root sign 5 / 5. (1) find the value of COS (α - β) (2) if - π / 2 < β < 0 < α < 2 / π, and sin β = - 5 / 13, find the value of sin α

It is known that I is a mathematical unit, if the complex number Z1 = cos α + isin α. Z2 = cos β + isin β satisfies | z1-z2 | 2 times the root sign 5 / 5. (1) find the value of COS (α - β) (2) if - π / 2 < β < 0 < α < 2 / π, and sin β = - 5 / 13, find the value of sin α

(1) | z1-z2 | = 2 √ 5 / 5 is carried into the complex number and sorted out as √ [(COS α - cos β) (COS α - cos β) + (sin α - sin β) (sin α - sin β)] = 2 √ 5 / 5
Using cos α cos β + sin α sin β = cos (α - β), it is obvious that cos (α - β) = 3 / 5
(2) If - π / 2 ＜ β ＜ 0 ＜ α ＜ 2 / π, then α - β ∈ (0,0.5 π) (because cos (α - β) = 3 / 5 > 0)
So α = β + arccos3 / 5
Sin β = - 5 / 13, cos β = 12 / 13
So sin α = sin (β + arccos 3 / 5) = 0.6 * sin β + 0.4 * cos β = 33 / 65
Note: the expansions of trigonometric functions sin (x + y) and COS (X-Y) are used in the solution

Who can explain Euler formula

Proof of Euler formula by topological method
Try to prove the Euler formula about the number of faces, edges and vertices of polyhedron by topological method
Euler's formula: for any polyhedron (that is, a solid whose faces are planar polygons and have no holes), suppose that f, e and V represent the number of faces, edges (or edges) and corners (or tops) respectively
F-E+V=2.
The proof is shown in Figure 15 (the figure is a cube, but the proof is general and "topological"):
(1) A polyhedron (1) is regarded as a hollow solid with a thin rubber surface
(2) If we remove one face of the polyhedron, we can completely spread it on the plane and get a straight line in the plane, as shown in Figure 2. Suppose that f ', e' and V 'respectively represent the number of (simple) polygons, edges and vertices of the plane figure, we only need to prove that f' - E '+ V' = 1
(3) For this plane figure, divide it into triangles. That is to say, for polygons that are not triangles, introduce diagonals one after another until they become triangles, as shown in Figure 3. For each diagonal introduced, f 'and E' increase by 1, while V 'does not change, so f' - e '+ V' does not change. Therefore, when it is completely divided into triangles, f 'and E' increase by 1, The value of F '- E' + V 'remains unchanged. Some triangles have one or two sides on the boundary of a plane figure
(4) If a triangle has one side on the boundary, such as △ ABC in Figure 4, remove the side of the triangle that does not belong to other triangles, that is, AC, so △ ABC is removed. In this way, f 'and E' are subtracted by 1 and V 'is unchanged, so f' - E '+ V' is not changed
(5) If a triangle has two sides on the boundary, such as △ def in Figure 5, remove the sides of the triangle that do not belong to other triangles, namely DF and EF, so △ DEF is removed. In this way, f 'minus 1, e' minus 2, V 'minus 1, so f' - E '+ V' remains unchanged
(6) Then f ′ = 1, e ′ = 3, V ′ = 3, so f ′ - e ′ + V ′ = 1-3 + 3 = 1
(7) Because the original figures are connected together, and the changes introduced in the middle do not destroy this fact, the final figures are connected together, so they will not be scattered in several triangles outward, as shown in Figure 7
(8) If we end up as shown in Figure 8, we can remove one of the triangles, that is, one triangle, three sides and two vertices. Therefore, f '- E' + V 'remains unchanged
That is f ′ - e ′ + V ′ = 1
So Euler's formula:
F-E+V=2
It has been proved
Or the following website has courseware to download

Is the complex impedance the value of the real part or the modulus of the vector?
Z = Z '+ JZ ", is the size of Z' or (Z '^ 2 + Z" ^ 2) ^ 0.5 at a certain time
In other words, is the actual value the value of the real part or the modulus of the vector?
I never understood the meaning of the imaginary part

I guess it's the latter. To be exact, any "vector" has no size, so it's not mathematically accurate to say the size of "impedance", but electricity may refer to its modulus

Solubility of junior high school chemistry
The following statement about solution is wrong
The mass fraction of solute in a-saturated solution may remain unchanged after crystallization
B chemical reactions in solution are usually faster
The mass fraction of solute in saturated solution of the same solute must be larger than that in unsaturated solution of C
The mass fraction of solute may remain unchanged when D unsaturated solution is converted to saturated solution
Help me analyze the right and wrong options
Another one is that the solubility of a solute does not change much with the increase of temperature, but the method of obtaining solute from its solution can usually be the method of evaporation solvent crystallization

The results show that a is correct, the crystal precipitated from saturated solution is still saturated solution, the mass fraction remains unchanged, B is correct, C is wrong, the concentration of saturated solution varies with temperature. The mass fraction of KNO3 saturated solution at 20 ℃ is not larger than that of KNO3 unsaturated solution at 70 ℃. D is correct, the concentration of saturated solution varies with temperature

In RLC series AC circuit, if inductance L = 1H and capacitance C = 1F are known, the frequency of resonance is F0 = () Hz
And explain what happens

F = 1 / (2 * pi * (L * C radical))

Achievements in 30 years of reform and opening up
100 words

In the past 30 years, unprecedented historical changes have taken place in all aspects of Chinese society. China's economic, political, cultural and social construction has made unprecedented historical progress

Plural forms of English words and some exercises,
human______
German______
Japanese______
American______
Three baskets of vegetables______
Eight pieces of wood______
Nine pieces of metal______
Two pieces of ice______
Six bowls of rice______
He began to work there _________ .
A.on his fifty
B.at age of fifty
C.when he fifty
D.in his fifties
(it seems that some of the options of this question are incomplete, but that's what's on the exercise paper.)
Jumping out of_______ airplane at ten thousand feet is quite______ exciting experience.
A.\,the
B.\,an
C.an,an
D.the,the

human-humans german-germans japanese-japanese american-americans3baskets of vegetalble 8pieces of wood 9 pieces of metal 2pieces of ice 6 bowls of rice D B

Find the surface area of the cylinder below: 1. The radius of the bottom is 3cm, the height is 5cm; 2. The perimeter of the bottom is 15.7cm, the height is 4cm; 3. The diameter of the bottom is 4cm, the height is 4cm
4. Side area 62.8 square decimeters, height 5DM

1.S=2*3.14*3^2+2*3.14*3*5=150.72(cm2)
2.r=15.7/(2*3.14)=2.5(cm)
S=2*3.14*2.5^2+15.7*4=102.05(cm2)
3.S=2*3.14*(4/2)^2+3.14*4*4=75.36(cm2)
4.C=62.8/5=12.56(dm)
r=12.56/(2*3.14)=2(dm)
S=2*3.14*2^2+62.8=87.92(dm2)

How to write the greatest common factor and the least common multiple of 26 and 182
plus

Greatest common factor 26
Least common multiple 182

In known isosceles triangle ABC, ab = AC, BD is the center line on the side of AC, BD divides the perimeter of triangle ABC into 9 and 12 parts, and calculates the side length of triangle

Because D is the midpoint of AC, and ab = AC, ad = AC = 1 / 2Ab. Then, let AB be 2x, BC be y, then ad and DC be X. ① when AB + ad = 9, DC + BC = 12: 2x + x = 9, x + y = 12, the solution is x = 3, y = 9. ② when AB + ad = 12, DC + BC = 9: 2x + x = 12, x + y = 9, the solution is x = 4, y = 5, So AB = AC = 6, BC = 9 or AB = AC = 8, BC = 5