Graph behavior of trig functions

Graph: A smooth wave oscillating between -1 and 1, starting at the origin (0, 0).
Period: The distance before the pattern repeats. The period of sin⁡(x) 2pi, , meaning it repeats every 2π units along the x-axis.
Amplitude: The height from the middle line (the x-axis) to the peak. For sin⁡(x), the amplitude is 1 (the function oscillates between -1 and 1).
Range: The set of possible output values. For sin⁡(x) the range is [−1,1]. Therefore, the values of sine can never be less than -1 or greater than 1.

Key Points:

Starts at (0,0), reaches (1,1) at pi/2, reaches zero at pi, reaches -1 at 3pi/2, and returns to zero at 2pi.

Graph: Similar to sine, but it starts at the maximum value (0,1)and oscillates between 1 and -1.

Period: Like the sine function, the period is 2π.

Amplitude: The amplitude is also 1 for cosine (oscillates between 1 and -1)

Range: : The range of cos⁡(x) is also [−1,1]

Key Points: Starts at (0,1), reaches 0 at pi/2, reaches -1 at pi, reaches 0 at 3pi/2, and returns to 1 at 2pi.

Graph:Has vertical asymptotes and cycles through all real numbers

Period: π
Amplitude: Undefined (since the function has vertical asymptotes).
Range: (−∞,∞)
Key Points:
- The graph passes through the origin (0, 0).
- Has vertical asymptopes at pi/2 and 3 pi/2.

Function: csc⁡(x)= 1/sin(x)
Graph: Reciprocal of the sine function, with vertical asymptotes where sin⁡(x)=0
Period: 2π
Amplitude: Undefined.
Range: (-infinity, -1]U[1, infinity)
Key Points: has a vertical asymptote where sin(x)=0. Peaks at (pi/2,1) and throughs at (3pi/2, -1)

Function: cot⁡(x)=1/tan(x)
Graph: Reciprocal of the tangent function, with vertical asymptotes where tan(x)=0.
Period: π
Amplitude: Undefined.
Range: (−∞,∞)
Key Points:
- Has vertical asymptotes where tan⁡(x)=0 (e.g. x=0, pi, 2pi …)

Here’s a practice set to help you reinforce the concepts!