Graphs in Trigonometry
Remember the unit circle?
Using the coordinates of the points fill in the table below
Now plot all of these points on the graph below.
Guidelines for graphing all curves:
1. We are going to hold the graph paper sideways
2. For this example we only need the first quadrant. (You can tell by where the table starts. Here we started at
zero.)
3. If you notice, the numbers on the table are very large. They go up to 360º and could go further. Because of
this we are ALL going to use the same intervals for EVERY graph no matter the table.
For the X- AXIS : Starting at 0, every six boxes is 90º. (Each box is 15º)
For the Y-AXIS : Starting at 0, every 4 boxes is 1. (Each box is .25)
The sine curve is a continuous curve that passes through (0,0)
Now complete the table and graph the cosine curve.
The cosine curve is a continuous curve passing through (0,1)
General Equations:
y = a sin b x
y = a cos b x
**Notice a is the coefficient of sine or cosine and b is the coefficient of x.
a: amplitude = |a|
The amplitude tells you how high and low the curve
will go.
It gives the range of the graph and can be
stated as : -a ≤ y ≤ a
It gives the min and max value points
(state as a positive number)
b: frequency |b|
The frequency of a trigonometric function is the number of cycles it completes in a given interval.
(3600)
Complete sine curve
Complete cosine curve
Period (p): p =
Tells you how many degrees it takes to draw each curve.
Example: State the amplitude, period, frequency and range of each function.
1. y = 3 sin 2x
2. y = cos(-4)x
Practice.
State the amplitude, period, frequency and range of each function.
1. y = sin x
2. y = -2 sin (1/2) x
3. y = 2 cos 3x
Graphing ANY sine or cosine curve:
Example: Graph y = sin2x on the interval 0≤ x ≤ 360º
1. Start by labeling the amplitude, period, frequency and range
2. You must create your table BUT you have to know what angle value to count by.(It will not always
be 90º) To tell what you will count by, look at b.
If b ≤ 1 count by 90ºs
If b = 2 count by 45ºs
If b = 3 count by 30ºs
(You will only have to know the 3 b’s above)
3. Also remember if the interval is given in radians, you must put a row in your table of all the radian
values.
Let’s now create a table for the example above: y = sin2x
Graphing Trig Functions on the Calculator
1. Put the equation into y = (Remember fractions need parentheses)
2. Hit Zoom 7 for the trig graph (Notice that is shows you from -360º to 360º so you may only be
graphing half of it on your graph paper)
3. To see the table with the values you want you must hit 2nd Window (Tbl set)
You want Tbl Start = 0 (Because your chart starts at 0)
You must change Δ Tbl = 30, 45, or 90 (Depending on what you are counting by)
4. Now you can hit 2nd Graph and all of your table values should be there.
Now plot the points from the table above on your axes and LABEL!
Do Now: Graph y = (1/2)cos 3x on the interval -π ≤ x ≤ π
(Notice here that the interval is in radians. Therefore we must put an additional radian row into our table. You
can convert to degrees first to see what numbers we are working with)
Practice: Graph each function on the given interval.
1. y
= 2 cos x on the interval 0≤x≤360
2.
y = sin 2x on the interval -180≤x≤180
3.
y = 3 cos 2x on the interval 0≤x≤2π
Systems of Trig Graphs
When graphing a system of equations, set up a different table for each equation. Graph both equations on the
same set of axes. Answer any questions that follow.
Ex:
1. a) On the same set of axes, sketch y = 2cosx and y = 1/2sinx on the interval 0<x<360.
b) Determine the number of values for which 2cosx = 1/2sinx.
2. a) On the same set of axes, sketch the graphs of y = 2sinx and y = cos1/2x on the interval 0<x<360.
b) Determine the number of values in which 2sinx = cos1/2x.
3. a) On the same set of axes, sketch the graphs of y = sinx and y = cosx on the interval -180<x<180.
b) Determine the values of x for which sinx = cosx.
Graphing Tangent
Find the value of each of the following:
tan 0 =
tan 90 =
tan 180 =
tan 270 =
tan 360 =
If we made a table of only these values we would have nothing to graph (zeros and undefined)
**Every time you graph tan, you MUST count by 45o**
1. Graph y = tan x from 0 < x < 360
Important info about the tan graph:
1. There are no straight lines
2. You must represent the asymptotes with a dotted line
3. Tan is a discontinuous curve
4. The period is always 180o
5. Always make the tan table counting by 45's
PHASE SHIFT
Phase shift is the measure horizontal shifting. If the phase shift is positive then the horizontal shift is to the
right. If the phase shift is negative, then the horizontal shift is to the left.
When determining the direction of the phase shift, it is important to have the function in the proper form:
y = Asin (B(x - C) + D
1.
2.
A) What do you notice about the shape of each graph?
3.
4.
A) What is happening to the graph?
B) What happens when you graph h(x) = sinx – 2?
In general:
Recall:
|A| = amplititude
B = frequency (cycles between 0 and 2π)
C = horizontal shift
D = vertical shift
When the value of D is positive, then the trigonometric graph shifts UP.
When the value of D is negative, then the trigonometric graph shifts DOWN.
Graphing Inverse Trig Functions
y = arcsinx = sin-1x
y = arccosx = cos-1x
y = arctanx = tan-1x
You cannot enter these functions into y = on your calculator to get a table.
In order to graph these functions by using a table:
1. Create a table for the original function
(i.e. if you want to graph y = arcsinx, then create a table for y = sinx)
2. Create a new table for y = arcsinx by switching the values of x and y from the other table.
3. Plot all points from table, remember 90o is now on the y-axis and 1 is on the x-axis!
1. Graph the function y = arcsinx
2. Graph the function y = Arccosx
Domain and Range of Inverse Graphs
Inverse Notation:
arcsin(x) = sin-1(x)
arccos(x) = cos-1(x)
arctan(x) = tan-1(x)
Caution!
As stated above, the exponent of -1
denotes "inverse". It does not mean
1/sin x, such as x-1 = 1/x
y = arcsin(x) = sin-1(x) solves the equation x = sin(y).
Read: arcsin(x) as "the angle whose sine is x".
Look at the sine function (in red) below.
If we reflect this function over the identity line, y = x
we will create the inverse graph (in blue).
Unfortunately, this newly formed inverse graph
is not a function. Notice how the green vertical
line intersects the new inverse graph in more
than one location, telling us it is not a function.
(Vertical Line Test).
By limiting the range to see all of the y-values without repetition, we can define inverse functions of the
trigonometric functions. It is possible to form inverse functions at many different locations along the graph. The
functions shown here are what are referred to as the "principal" functions.
Inverse sine:
f(x) = sin-1(x)
f(x) = arcsin(x)
Domain: [-1,1] or {-1 < x < 1}
Range: [-π/2, π/2] or {-π/2 < x < π/2}
Inverse cosine:
f(x) = cos-1(x)
f(x) = arccos(x)
Domain: [-1,1] or {-1 < x < 1}
Range: [0, π] or {0 < x < π}
Inverse tangent:
f(x) = tan-1(x)
f(x) = arctan(x)
Domain: [-∞, ∞] or {-∞ < x < ∞} or {All Real Numbers}
Range: (-π/2, π/2) or {-π/2 < x < π/2}
Multiple Choice.
1. Find the value of θ in radians considering
2. Which equation is equivalent to
the principal inverse function: θ = arcsin(1)
y = sin-1(x)? {-π/2 < x < π/2}
3. The inverse of Cos x is a function. The domain of Cos x could be
(1) {x: -2π < x < 0}
(2) {x: 0 < x < 2π}
(3) {x: -π < x < π}
(4) {x: 0 < x < π}
4) If Arcsin is a function, the domain of Sin x could be
(1) {x: -2π < x < 0}
(2) {x: -π/2 < x < π/2}
(3) {x: -π < x < π}
(4) {x: 0 < x < 2π}
5) Tan-1x is a function. The domain of Tan x could be
(1) {x: -2π < x < 0}
(2) {x: -π/2 < x < π/2}
(3) {x: -π/2 < x < π} (4) {x: -π < x < π}