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Graphs of the Circular Functions

This page is meant to serve as a quick overview of the graphs of the circular functions.

Cosine and Sine Functions

y = cos x y = sin x
Domain: (-∞,∞) Domain: (-∞,∞)
Range: [-1,1] Range: [-1,1]
Amplitude: 1 Amplitude: 1
Period: 2π Period: 2π

Basic format of sine and cosine graph equations and what can be derived from them.

y = c + a sin b(x-d) or y = c + a cos b(x-d)
Amplitude: |a| Period: 2π / b
Verticle Translation: c units. Phase Shift: d units.

Instructions to graphing the function.

Step 1 Find the information that can be derived from the equation, using the information provided above.
Step 2 Using your knowledge of a basic sine or cosine wave as viewable above, stretch or compress the graph as necessary to match the period and amplitude.
Step 3 Shift the graph according the the verticle translation and the phase shift as found in Step 1.
And now you have the graph of the sine or cosine function.

Secant and Cosecant Functions

y = sec x y = csc x
Domain: {x|x ≠ π/2 + nπ} where n is any integer. Domain: {x|x ≠ π/2 + nπ} where n is any integer.
Range: (-∞,-1] U [1,∞) Range: (-∞,-1] U [1,∞)
Amplitude: DNE Amplitude: DNE
Period: 2π Period: 2π

Instructions to graphing the function. These instructions apply to secant and cosecant equations that are in the form y = a csc bx or y = a sec bx

Step 1 Graph the reciprocal function to use as a guide. Note: It is recommended that this is lightly sketched or drawn using dashed lines.
Step 2 Draw in the verticle asymptotes. Note: It is recommended that this is lightly sketched or drawn using dashed lines.
Step 3 Sketch in the graph of the function, using the graph drawn in step 1 and the boundries drawn in step 2 as guides. It should resmemble the images shown above for the secant and cosecant graphs.
And now you have a sketch of the secant or cosecant function.

Tangent and Cotangent Functions

y = tan x y = cot x
Domain: {x|x ≠ π/2 + nπ} where n is any integer. Domain: {x|x ≠ π/2 + nπ} where n is any integer.
Range: (-∞,∞) Range: (-∞,∞)
Amplitude: DNE Amplitude: DNE
Period: π Period: π