Graphs of Sine and Cosine

Graphs

y = sin x   for   0 < x < 2p
Domain:  all real numbers
Range:  -1 < y < 1
Period: 2p

y = cos x    for     0 < x < 2p
Domain:  all real numbers
Range:  -1 < y < 1
Period: 2p
 

y = tan x   for    -p/2 < x < p/2
Domain:   x is all real numbers except p/2 + np  (asymptotes occur at these values)
Range:  all real numbers
Period:  p
 

y=cot x  for   0 < x < p
Domain:  all real numbers except np  (asymptotes occur here)
Range:  all real numbers
Period:  p

y = sec x   for    0 < x < 2p
Domain:   x is all real numbers except p/2 + np  (asymptotes occur at these values)
Range:  y > 1, y < -1
Period:  2p

y = csc x    for    0 < x < 2p
Domain:  all real numbers except np  (asymptotes occur here)
Range:  y > 1, y < -1
Period:  2p

Amplitude

For   y = A sin B(x - C) + D,    |A| is the amplitude.

amplitude =   |Max - Min|
                             2

Period

For    y = A sin B(x - C) + D,  2p / |B|   is the period.

|B| is the number of times the sine curve repeats itself for 0 < x < 2p

Horizontal Shift

For     y = A sin B(x - C) + D, C is the horizontal shift.  Solve  x - C = 0 for x to see where to start the new period.

Vertical Shift

For    y = A sin B(x - C) + D,   D is the vertical shift.  This is the amount the curve is moved up or down from the x-axis.

Equations

Solve:   sin 2x = 1/2  for   0 < x < 2p

Since   0 < x < 2p, then   0 < 2x < 4p

                  2x = sin-1(1/2)
                  2x = p/6, 5p/6, 13p/6, 17p/6 (all angles between 0 and 2p)
                 x = p/12, 5p/12, 13p/12, 17p/12 (dividing both sides by 2)

Problems

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