Final answer:
To sketch the graphs of the trigonometric functions, we consider their behavior within a given interval. Each function has specific characteristics that determine its graph. For cos x and sin x, we plot points at various x-values and connect them smoothly. For tan x and csc x, we need to consider their asymptotes and the repeated cycles of their values.
Step-by-step explanation:
To sketch the graph of each function, we need to consider the behavior of the functions within the given interval of [-2π,2π]. Here are the steps to sketch each function:
2. y=cos x
The graph of y = cos x is a periodic function that oscillates between -1 and 1. It completes one cycle (from 0 to 2π) in 2π radians. To sketch the graph, plot points at various x-values and connect them smoothly. Label the x-coordinates of the zeros which occur at x = π/2 and x = 3π/2.
3. y=tan x
The graph of y = tan x is a periodic function that has vertical asymptotes at x = π/2, 3π/2, 5π/2, etc. It has repeated cycles of increasing and decreasing values. To sketch the graph, plot points at various x-values and connect them smoothly, making sure to avoid the vertical asymptotes.
4. y=sin x
The graph of y = sin x is a periodic function that oscillates between -1 and 1. It completes one cycle (from 0 to 2π) in 2π radians. To sketch the graph, plot points at various x-values and connect them smoothly. Label the x-coordinates of the zeros which occur at x = 0, π, 2π, etc.
5. y=csc x
The graph of y = csc x is the reciprocal of the graph of y = sin x. It has vertical asymptotes at x = 0, π, 2π, etc. To sketch the graph, plot points at various x-values and connect them smoothly, making sure to avoid the vertical asymptotes.