In the same rectangular coordinate system, there are several intersections between the image of function y = SiNx and the image of y = X

In the same rectangular coordinate system, there are several intersections between the image of function y = SiNx and the image of y = X

Hello! First of all, according to the general image, the intersection point is at (- π / 2, π / 2) and obviously (0,0) is the intersection point. Because both of them are odd functions, we can prove that x > SiNx in (0, π / 2) by definition, as shown in the figure, in the unit circle, x = the length of arc AB / ob = arc ab

In the same coordinate system, the image of function y = SiNx and the image of function y = x have several common points I think there are three, one in the first quadrant, one in the origin, and one in the third quadrant

This can be solved by using the derivative in high school mathematics. Y '= cosx, y' represents the slope of y = SiNx. It can be seen that it is always smaller than 1, so y = x and y = SiNx cannot have intersection points in the first and third quadrants

Shift the image of function y = sin4x to the left π If we get the image of y = sin (4x + φ), then φ is equal to () A. -π Twelve B. -π Three C. π Three D. π Twelve

The image of the function y = sin4x shifts π to the left
12 units, y = SiN4 (x + π)
12) The image of y = sin (4x + φ), so φ = π
Three
Therefore, C is selected

The image of the function y = sin (2x + π / 4) is shifted to the right by π / 8 units, and then the abscissa is compressed to the original 1 / 2 The analytic expression () a, y = sin (4x + π / 8) B, y = sin (4x + π / 32) [detailed process, thank you] process

Is there only two answers? It should be y = sin4x

Let y = sin (2x + 5 π) 6) At least shift the image to the left______ The graph of an even function can be obtained

Let y = sin (2x + 5 π)
6) The image is shifted to the left by π
Three units give y = sin [2 (x + π)
3）+5π
6]=sin2（x+3π
2 ）=-cos2x，
And y = - cos2x is an even function,
So the answer is: π
3．

If the image of the function y = sin (2x + φ) is shifted to the left by π / 8 units along the axis, an image of even function is obtained, then one of the possible values of φ is () The answer is π / 4. I calculated 3 π / 4. Here is my method. What's wrong? Y=sin(2x+φ) Y=sin2(x+φ/2） After translation: y = sin2 (x + φ / 2 + π / 8) =sin2(x+(4φ+π)/8) So (4 φ + π) / 8 = 1 / 2 π + K π φ=3π/4+2kπ

Your mistake is that sin2 (x +...) does not multiply "2" by what is in brackets
It should be: y = sin [2x + 2 * (4 φ + π) / 8] = sin (2x + (4 φ + π) / 4]
If y is even, then (4 φ + π) / 4 = 2K π + π / 2
∴φ=2kπ+π/4,k∈Z.
One possible value of φ is φ = π / 4

If the image of function y = sin (2x + π / 6) is shifted n units to the left, the corresponding function of the graph is even function, then the minimum value of n is () I calculated - π / 3, and the answer was π / 12 Let y = sin2 (x + π / 12 + n) = sin (2x + π / 6 + 2n) be the minimum value when π / 6 + 2n = - π / 2?

No. the minimum value should be such that y = sin (2 (x + n) + π / 6) = sin (2x + π / 2 + K π) (k is an integer), 2n + π / 6 = π / 2 + K π, n = 1 / 6 π + (1 / 2) k π, when k = 0, the minimum value of n is 1 / 6 π

To get the image of the function y = cos2x, simply change the function y = sin (2x - π) 3) Image of () A. Shift 5 π to the left Six B. Shift right 5 π Six C. Shift 5 π to the left Twelve D. Shift right 5 π Twelve

∵y=cos2x=sin（2x+π
2）
Let's suppose that we only need to change the function y = sin (2x - π)
3) Then, the image is translated by φ units
sin[2（x+φ）-π
3]=sin（2x+π
2）
∴2（x+φ）-π
3=2x+π
2，φ=5π
Twelve
Therefore, we should shift 5 π to the left
12 units
Therefore, C

To get the image of function y = cos2x, how to translate the image of y = sin 2x?

Because cos2x = sin (2x + π / 2) = sin2 (x + π / 4)
To get the image of the function y = cos2x, just take the image of y = sin 2x
Shift π / 4 units to the left

How to transform the image of function y = SiNx to obtain y = sin (1 / 2x + π / 3),

Firstly, π / 3 is shifted to the left, and then the x-axis is extended to 2 times
Or the x-axis is lengthened by 2 times and then shifted to the left by π / 6