It is proved that sin (x + y) sin (X-Y) = SiNx siny

It is proved that sin (x + y) sin (X-Y) = SiNx siny

sin(x+y)sin(x-y) =-1/2(cos(x+y+x-y)—cos(x+y-x+y)) =-1/2(cos2x—cos2y) =-1/2(1-2（sinx）^2-1+2(siny）^2) =（sinx）^2-(siny）^2

How to prove SiNx + siny = 2 * sin (x + Y / 2) * cos (X-Y / 2)

Let a = (x + y) / 2, B = (X-Y) / 2
X=A+B,Y=A-B
SINX=SIN(A+B)=SINACOSB+COSASINB
SINY=SIN(A-B)=SINACOSB-COSASINB
SINX+SINY=2SINACOSB

It is proved that SiNx + siny + Sinz sin (x + y + Z) = 4sin ((x + y) / 2) sin ((x + y) / 2) sin ((x + y) / 2)

sinx+siny+sinz-sin(x+y+z)=4sin[(x+y)/2]sin[(x+z)/2]sin[(y+z)/2]
sinx+siny+sinz-sin(x+y+z)
=2sin[(x+y)/2]cos[(x-y)/2]+sinz-sin(x+y)cosz-sinzcos(x+y)
=2sin[(x+y)/2]cos[(x-y)/2]+sinz[1-cos(x+y)]-sin(x+y)cosz
=2sin[(x+y)/2]cos[(x-y)/2]+2sinz*sin[(x+y)/2]^2-2sin[(x+y)/2]cos[(x+y)/2]cosz
=2sin[(x+y)/2]*{cos[(x-y)/2]+sinzsin[(x+y)/2]-cos[(x+y)/2]cosz}
=2sin[(x+y)/2]*{cos[(x-y)/2]-cos[z+(x+y)/2]}
=2sin[(x+y)/2]*2sin[(x+z)/2]sin[(y+z)/2]
=4sin[(x+y)/2]sin[(x+z)/2]sin[(y+z)/2]

The process of proving sin (x + y) = SiNx * cosy + cosx * siny Thank you for the proof process

This paper recommends a vector method to prove that:
Take two points m n on the unit circle, and the angles with the X axis are x, PI / 2 + y, respectively
Then M (cosx, SiNx), n (- siny, cosy)
（OM,ON）=
Cos (OM, on) = cos (PI / 2 + Y-X) = sin (X-Y) = om dot multiplication on / (| om | on |)
In other words, it is SiNx Syx
By replacing y with - y, we can get: sin (x + y) = sinxcosy + cosxsiny
Conclusion
Similarly, the formula of COS (x + y) can be obtained

If the image of function y = sin (2x + π / 3) is translated by vector a, the image is symmetric about the center of point (- π / 12,0), then the coordinate of vector a is If (a * 2 + C * 2 + b * 2) tanb = radical 3aC, find the angle B

1. K π / 2 - π / 12 K is the natural number
2.=-3

If the image of function y = sin (2x + π / 3) is translated according to vector φ, the image obtained is symmetrical about the straight line x = π / 6, then the coordinates of vector φ may be (,) If the image of function y = sin (2x + π / 3) is translated according to vector φ, the image obtained is symmetrical about the straight line x = π / 6, then the coordinates of vector φ may be (,)

If the image of function y = sin (2x + π / 3) is translated according to vector φ, the image obtained is symmetrical about the straight line x = π / 6, then the coordinates of vector φ may be (,) analytic:

After the image of function y = sin (2x + (PIE) / 3) is translated by vector a, the image obtained is symmetric about (- (PIE) / 12,0) After the image of function y = sin (2x + (PIE) / 3) is translated by vector a, the image obtained is symmetric about (- (PIE) / 12,0), and the vector a (where pie refers to a mathematical symbol, originally 3.14159..., in this case, is the radian system)

Let a = (n, 0) translate y = sin [2 (x-n) + π / 3] = sin (2x-2n + π / 3)
The point (- π / 12,0) is symmetric, so sin [2 (- π / 12) - 2n + π / 3] = sin (- 2n + π / 6) = 0
So - 2n + π / 6 = k π, that is, n = - K π / 2 + π / 12
So when k = 0, the absolute value of n is the smallest, so a = (π / 12,0)

Translation of trigonometric functions of higher one How to translate the y = sin (2x + π / 3) image into y = sin (2x + π / 3) image?

Y = sin (2x + π / 3) can be converted into sin2 (x + π / 6). According to the principle of left plus right subtraction, it is obvious that the image moves π / 6 along the negative direction of X axis

There is a function y = sin (2x - π / 3) with a function y = sin (2x - π / 3). When you want to shift it to the right and shift it to the units of π / 3, why should we first separate the coefficient of the function into y = sin2 (x - π / 6 - π / 6) and then change y = sin2 (x - π / 6 - π / 3) to solve? When the period of the function y = sin (2x - π / 3) is reduced to half of the original, when the period of the function y = sin (2x - π / 3), directly add 4 before x before x, change 4 to y = sin (4x - 2x - π / 3 (4x - π / 3) instead of looking X - π / 6 as a whole, turning X - π / 6 3) what about it? Is it possible to change y = sin (Wx + φ) into y = SiNW (x + φ / W) when it comes to periodic transformation, and then find it according to the meaning of the question?

Yes, the translation transformation and the expansion transformation are for "one X". You can do these problems by taking special values, which may be easier to understand. For example, when x = π / 3, y takes the maximum, then the right translation of π / 3 should be x = π / 3 + π / 3 = 2 π / 3. You can see whether it is right or not

The function y = The monotone decreasing interval of 2x2 − 3x + 1 is______ .

Let t = 2x2-3x + 1 and t ≥ 0
Its symmetry axis is: x = 3
4, and X ∈ (− ∞, 1)
2]∪[ 1，+∞)
The monotone decreasing interval of T is (− ∞, 1
2]
And ∵ y =
T is an increasing function on [0, + ∞),
The function y =
The monotone decreasing interval of 2x2 − 3x + 1 is (− ∞, 1)
2]
So the answer is: (− ∞, 1
2]