At what angle to the x-axis does the graph of y=sin x pass through (0,0)? Give numerical and visual evidence.
Hint: it is zero degrees. PLot it.
The angle would be the same as a tangent to the curve at that point
dy/dx = cosx
at (0,0) , dy/dx = cos0 = 1
dy/dx = 1 = tanØ, where Ø is the angle it makes with the x-axis
tanØ = 1
Ø = 45° or π/4 radians
To determine the angle at which the graph of y = sin(x) passes through the point (0,0), we need to find the value(s) of x that satisfy this condition.
Analytically, we can observe that the point (0,0) lies on the x-axis. Therefore, y must be equal to 0. Substituting y = 0 into the equation y = sin(x) gives us:
0 = sin(x)
Now, let's determine the values of x that satisfy this equation. Recall that sin(x) = 0 when x is an integer multiple of π. In other words, x can take on values such as 0, π, 2π, -π, -2π, etc.
To provide numerical evidence, we can calculate the values of x using this information. By starting with 0 and adding or subtracting multiples of π, we get the following values for x:
x = 0, π, -π, 2π, -2π, ...
Hence, the graph of y = sin(x) passes through the point (0,0) at x = 0, π, -π, 2π, -2π, and so on.
For visual evidence, we can plot the graph of y = sin(x) and observe where it intersects the x-axis. By using graphing software or calculators, we can plot the function and see that it crosses the x-axis at x = 0, π, -π, 2π, -2π, as shown in the graph below:
[Graph]
Therefore, both numerical and visual evidence support the conclusion that the graph of y = sin(x) passes through the point (0,0) at x = 0, π, -π, 2π, -2π, and so on.