On the graph of y = sin x, many points have 0.39073 as the y-coordinate, among them P = (157, 0.39073). Without relying on a calculator or graphing tool, show how you can determine the coordinates of the two closest to P.
To determine the coordinates of the points closest to P=(157, 0.39073) on the graph of y = sin x without relying on a calculator or graphing tool, we can use some properties of the sine function.
1. The graph of y = sin x repeats itself every 2π units in the x-direction. This means that if (x, y) is a point on the graph, then (x + 2π, y) is also a point on the graph.
2. The sine function is symmetric about the origin. This means that if (x, y) is a point on the graph, then (-x, -y) is also a point on the graph.
Since P = (157, 0.39073) lies in the positive y-region, we can determine the coordinates of the two points closest to P as follows:
1. Using property 1 above, we can find the point Q = (157 + 2π, 0.39073), which is also on the graph.
2. Using property 2 above, we can find the point R = (-157, -0.39073), which is also on the graph.
Therefore, the coordinates of the two points closest to P are Q = (157 + 2π, 0.39073) and R = (-157, -0.39073).
To determine the coordinates of the two points on the graph of y = sin x that are closest to P = (157, 0.39073), we can follow these steps:
Step 1: Find the period of the sine function:
The period of the sine function is equal to 2π. This means that the graph of y = sin x repeats itself every 2π units horizontally.
Step 2: Find the phase shift of the sine function:
The phase shift of the sine function is equal to 0, which indicates that the graph is not shifted left or right.
Step 3: Determine the horizontal position of P within one period:
Since the period is 2π, and P has an x-coordinate of 157, we need to determine the horizontal position of P within one period of the sine function. To do this, we can divide the x-coordinate of P by the period (2π) and take the remainder:
Horizontal position of P within one period = 157 mod (2π)
Step 4: Determine the closest points:
To find the two points on the graph closest to P, we need to consider the points on the graph that have y-coordinates closest to 0.39073. Since sine oscillates between -1 and 1, we can use the fact that the sine function is symmetrical about the x-axis to find the closest points.
The two closest points to P will be symmetrical about the x-axis and have y-coordinates of 0.39073. We can find these points by finding the x-values that are symmetrical to the x-coordinate of P about the vertical line x = 0.
Step 5: Determine the coordinates of the two closest points:
To find the coordinates of the two closest points on the graph, we can use the x-values obtained in Step 4 and substitute them into the equation y = sin x.
Therefore, the coordinates of the two points closest to P = (157, 0.39073) can be found by substituting the x-values into the equation y = sin x, considering the symmetry about the x-axis.
To determine the coordinates of the two points closest to P = (157, 0.39073), we can use the properties of the sine function.
1) First, we need to find the period of the sine function. The period of the sine function is the distance between two consecutive peaks or troughs, which is equal to 2π (or approximately 6.28). Since the given equation is y = sin x, we know that the period is 2π.
2) Next, we need to find the phase shift or horizontal shift of the function. The sine function is centered around the origin (0,0) and starts at (0,0) for x = 0. However, the given equation is y = sin x, not y = sin(x - a) with some phase shift. Therefore, there is no phase shift, and the function starts at (0,0).
3) Now, we can calculate the x-coordinate of the closest points. Since the period of the sine function is 2π, we know that one complete cycle occurs every 2π units. Therefore, to find the x-coordinate of the two closest points to P, we need to find the values of x that are the closest to 157 and are separated by a multiple of 2π.
Let's assume one of the closest x-coordinates is x1. To find x1, we can subtract or add a multiple of 2π from 157 until we get the closest value that is less than or equal to 157.
Starting with the calculation for x1:
x1 = 157
x1 = 157 - 2π
x1 ≈ 157 - 6.28
x1 ≈ 150.72
Therefore, the x-coordinate of one of the points closest to P is approximately 150.72.
Similarly, we can find the other x-coordinate, x2, by adding another multiple of 2π to 157:
x2 = 157 + 2π
x2 ≈ 157 + 6.28
x2 ≈ 163.28
So, the x-coordinate of the other point closest to P is approximately 163.28.
To find the corresponding y-coordinates, we can substitute these x-coordinates into the equation y = sin x:
For the first point (x1 ≈ 150.72):
y1 = sin x1 ≈ sin 150.72 ≈ 0.39073
For the second point (x2 ≈ 163.28):
y2 = sin x2 ≈ sin 163.28 ≈ 0.39073
Therefore, the two closest points to P = (157, 0.39073) are approximately (150.72, 0.39073) and (163.28, 0.39073).