7.1 Graphs of the Sine, Cosine, and Tangent Functions
 graphs of trig. Functions are directly related to angles in the unit circle
#1. Graphs of Sine Function – think how the values of sine change as you go from
interval to interval
a) 0 to π/2 – increasing from 0 to 1
b) π/2 to π – decreasing from 1 to 0
c) π to 3π/2 – decreasing from 0 to -1
d) 3π/2 to 2π – increasing from -1 to 0
** if you follow the graph further out  to 4π you will see that the curve retraces
the path taken from 0 to 2π
** Period = sin(t ± 2kπ) = sin t
where k is any integer
#2. Graphs of cosine function
a) 0 to π/2 – decreasing from 1 to 0
b) π/2 to π – decreasing from 0 to -1
c) π to 3π/2 – increasing from -1 to 0
d) 3π/2 to 2π – increasing from 0 to 1
** just like the sine function, cosine retraces path at intervals of 2π
** Period = cos(t±2kπ) = cos t
where k is any integer
**The domain of the sine and cosine functions is the set of all real #’s **
 the y-coordinate of every point on these graphs lies between -1 and 1 so that the
range of the sine and cosine functions is the interval [-1,1]
#3. Graph of the Tangent Function
 the graph of f(t)=tan t can be sketched by noting the slope of the terminal side of
an angle of t radians
Change in t
0 to π/2
Movement of terminal side
-from horizontal upward toward
Vertical
terminal side slope
-increases from 0 in the
positive direction
0 to –π/2
-from horizontal downward
Toward vertical
-decreases from 0 in the
negative direction
**The graph of a tangent repeats this pattern at intervals of length π **
DOMAIN: is all real #’s except odd multiples of π/2
RANGE: all real #’s
PERIOD: tan(t±kπ) = tan t
**vertical asymptotes at π/2, -π/2 and 3π/2, -3π/2
**where k is any integer
Ex. graph g(x) = 2cosx  vertical stretch of cos x by factor of 2
Ex. -⅓sin t  reflection and compression of sin x by ⅓ and
Even Functions
 if f(-x) = f(x) for every x in the domain of f
** the graph of an even function is symmetric with respect to the y-axis
Ex. f(t) = cost is an even function because cos(-t) = cos(t) for every t in domain
Odd functions
 if f(-x) = -f(x) for every x in domain of f
**graph is symmetric with respect to the origin
Ex. f(t) = sin t and g(t) = tan t
odd because
Sin(-t) = -sint
tan(-t) = -tant
Sine
Domain
all reals
Range
[-1,1]
Period
2π
Even/Odd
Odd (symmetric to origin)
Cosine
all reals
[-1,1]
2π
Even (symmetric to y-axis)
Tan
all reals
all reals
Except odd
Intervals of π/2
π
Odd (symmetric to origin)
7.2 Graphs of Cosecant/Secant/Cotangent
#1. Cosecant – graph determined by using the graph of sine and the fact that csc t=1/sint
a) Graph: f(x) = sin x
g(x) = 1/sin x
b) Window: -2π<x<2π
-4 ≤ y ≤ 4
c) How alike: 1) + over same intervals
2) same period 2π
- over same intervals
e) How different: 1) every point where graph touches = local max for 1 graph,
local mine for the other graph
2) f is increasing on intervals where g is decreasing
3) f is concave down on intervals where y is concave up and
vice versa
**Because are reciprocals of each other
when x is an integer multiple of π because csc is not defined when
Sin x = 0 (ex. 3π,4π)
DOMAIN of CSC X = all real #’s except integer multiples of π, (vertical
Asymptotes at these points)
RANGE: all real #’s ≥ 1 or ≤ -1
PERIOD: 2π
Sin increases to height of 1 then  csc x decreases to height of 1
Sin  decreases to height of -1 then  csc x increases to height of 1
#2. Secant - related to the cosine function
a) graph f(x) = cos x
g(x) = 1/cos x
b)
Window: -2π≤x≤2π
-4≤y≤4
c) Alike: 1) positive over same intervals and negative over same intervals
2) same period of 2π
d) Different: 1) every point where touch  local max. on one and local min on
other
2) f is increasing on intervals where g is decreasing and vice versa
3) f is concave down on intervals where g is concave up and vice
versa
**Because are reciprocals – sec x is not defined when cos = 0 at odd multiples of π/2
DOMAIN OF SEC: all real #’s except of odd multiples of π/2 (π/2, 3π/2, 5π/2)
RANGE OF SEC: all real #’s such that x≥1 or x≤ -1
PERIOD: 2π
Ex.
#3. Cotangent = cos/sin
so graph is not defined when sin = 0
** sin t = 0 whenever t is an integer multiple of π
DOMAIN OF COT: all real #’s except integer multiples of π (2π, 3π, 4π)
RANGE OF COT: all real #’s
**vertical asymptotes at integer multiples of π **
**As graph of y = tan t increases, graph of cot t decreases and vice versa
PERIOD OF COT = π
Ex.
DOMAIN
*all reals except odd
Multiples of π/2
RANGE
all x≥1 or x≤-1
Csc
*all reals except
Multiples of π
all x≥1 or x≤-1
2π
odd
Cot
*all reals except
Multiples of π
all real #’s
π
odd
Sec
PERIOD
2π
EVEN/ODD
even
7.3 Period, Graphs and Amplitude
**all graphs of sine and cosine are periodic and graphs consist of a series of identical
waves
1 simple wave = cycle  length of each cycle is the period
For sin:
for cos:
PERIOD: when in form of g(x) = cosbx or f(x) = sinbx (where b is a constant)
**it changes the period of the sine or cosine  either by increasing or decreasing
The length of each cycle.
**If b>0, then graphs of either f(x) = sinbx or g(x)=cosbx
Makes b complete cycles between 0 and 2π and each function has a period of
2π
b
Ex. determine the period of each:
a) cos5t =
b)sint/2
PERIOD OF TANGENT: if b>0, then graph of f(x)=tanbx makes b complete cycles
between –π/2 and π/2 and the function has a period of π/b
Ex. determine the period of:
a) f(x) = tan5x
b) f(x)= tant/2
AMPLITUDE: If a≠0 and b>0 then each of the functions:
F(x)=asinbx
g(x)=acosbx
Has an amplitude of |a| and a period of 2π/b
Ex. find the amplitude and period of
a) f(t)= 3sint, then graph on interval of [-π,π]
Ex. Sketch a cycle of: 4cosx/2
7.4 Periodic Graphs and Phase Shifts
** In the last section, we saw how a and b constants affect amplitudes and periods of
F(x) = asinbx and g(x)= acosbx
**Now we will look at f(x)=asin(bx-c) +d and g(x)=acos(bx-c) + d
where a, b, c, d are all constants
Ex. Describe -2cost + 3 on interval [-2π, 2π]
(is the graph of cos t  reflected
across x-axis, vertically stretched
by a factor of 2, shifted up 3 units)
Phase Shifts – remember horizontal translations (x-c) where c is a constant
**in periodic functions, c = phase shifts
Ex. Describe the graph of each:
a) sin(t-¾π)
b) cos(t + π/4)
Ex. Graph the following by hand: f(t) = sin(t - ¾π)
Combined transformations :
Ex. State the amplitude, period, and phase shift of f(x) = 2cos(3x-4)
Summary of Combined Transformations:
If a≠0 and b>0 then each of the functions
F(x) = asin(bx-c) +d and g(x) = acos(bx-c) + d
Amplitude=|a|
Period = 2π
b
Phase Shift= c
b
vertical shift = d
Ex. graph 3sin(2t - 5π/4)
Graphs and Identities – a calculator can prove that a particular equation is not an identity
Ex. cos (π/2 + t) = sin t
so: f(t) = cos (π/2 + t) and g(t) = sin t
**if they are = then would have the same graph - put in calc.
Don’t so not identity
Ex. cos(π/2 – x) = sin x
same so are identities
Ex. sin t
Tan t = cos t
coincide – so are identities
7.4A
Other Trig. Graphs
Sinusoidal Graphs – if b, d, k, r, and s are constants, then graph of the function
G(t) = dsin(bt + r) + kcos(bt + s)
Is a sinusoid and there are constants a and c such that
Dsin(bt + r) + kcos(bt + s) = asin(bt + c)
Ex. Find the sine function whose graph looks like graph of: g(t)=3sin(2t + 1) +4cos(2t-3)
Steps: graph in calculator
Amplitude = 2.58 (use trace feature)
Period = π
Phase shift = .26 - -c/b = .26 (solve for c)