Compare the graphs
f(¦È)=sin¨¨ and f(¦È)=sin(-¦È).
f(¦È)=cos¨¨ and f(¦È)=cos(-¦È).
What conclusions can you make?
To compare the graphs f(θ) = sinθ and f(θ) = sin(-θ), and f(θ) = cosθ and f(θ) = cos(-θ), we need to understand the properties of these trigonometric functions and how they behave with respect to changes in the angle.
1. f(θ) = sinθ and f(θ) = sin(-θ):
The sine function is an odd function, which means it has symmetry about the origin. This property allows us to compare f(θ) = sinθ and f(θ) = sin(-θ).
To determine the relationship between the graphs, we can substitute values of θ as input and observe the corresponding values of sinθ and sin(-θ).
When we substitute positive values of θ (e.g., θ = 0, θ = π/2, θ = π), we will obtain the same values of sinθ for both f(θ) = sinθ and f(θ) = sin(-θ). This is because the sine function is periodic and repeats itself.
However, when we substitute negative values of θ (e.g., θ = -π/2, θ = -π), we will obtain the same magnitude of sinθ but with opposite signs between f(θ) = sinθ and f(θ) = sin(-θ). This means that the graph of f(θ) = sinθ is symmetric with respect to the origin, while the graph of f(θ) = sin(-θ) is symmetric with respect to the x-axis.
2. f(θ) = cosθ and f(θ) = cos(-θ):
The cosine function is an even function, which means it has symmetry about the y-axis. This property allows us to compare f(θ) = cosθ and f(θ) = cos(-θ).
Similarly, we can substitute values of θ as input and observe the corresponding values of cosθ and cos(-θ).
When we substitute positive values of θ (e.g., θ = 0, θ = π/2, θ = π), we will obtain the same values of cosθ for both f(θ) = cosθ and f(θ) = cos(-θ). This is because the cosine function is periodic and repeats itself.
As with the sine function, when we substitute negative values of θ (e.g., θ = -π/2, θ = -π), we will obtain the same magnitude of cosθ but with the same signs between f(θ) = cosθ and f(θ) = cos(-θ). This means that both graphs of f(θ) = cosθ and f(θ) = cos(-θ) are symmetric with respect to the y-axis.
In conclusion:
For the sine function, f(θ) = sinθ and f(θ) = sin(-θ) have symmetrical graphs about the origin.
For the cosine function, f(θ) = cosθ and f(θ) = cos(-θ) have symmetrical graphs about the y-axis.