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Unformatted text preview: Pre – Calculus Math 40S: Explained!
www.math40s.com 42 TRIGONOMETRY LESSON FIVE
PART I  PERIOD
The period of a graph is defined as the length of one complete cycle. This graph has a
period of 2π This graph has a
period of 4π This graph
has a period
of π
Most graphs given to
you won’t be as
simple as the first
three. In trig graphs
that are continuous,
you will have to first
identify a sine or
cosine pattern before
you can determine the
period.
The easiest way to do
this is to draw a
square around either
pattern and look at
the length.
The graph on the left
has a period of π Pre – Calculus Math 40S: Explained!
www.math40s.com 43 TRIGONOMETRY LESSON FIVE
PART I  PERIOD
Questions: For each of the following graphs, draw a rectangle around the indicated
pattern and state the period.
1) Draw a rectangle around a sine pattern. 2) Draw a rectangle around a cosine pattern. 3) Draw a rectangle around a sine pattern. 4) Draw a rectangle around a sine pattern. 5) 6) Draw a rectangle around a cosine pattern. Draw a rectangle around a sine pattern. Pre – Calculus Math 40S: Explained!
www.math40s.com 44 TRIGONOMETRY LESSON FIVE
PART I  PERIOD
ANSWERS: 1) Period = 8π 2) Period = 4π 3) Period = 4π 4) Period = 5) Period = π 6) Period = π
3 2π
3 Pre – Calculus Math 40S: Explained!
www.math40s.com 45 TRIGONOMETRY LESSON FIVE
PART II  THE B VALUE
The “b” value represents the number of cycles a trig graph has within a span of 2π.
It is the number that you see in a trig function right beside θ. (y = sinbθ)
The b value is NOT the period.
The bvalue and period (for radians) are related by the formula: The bvalue and period (for degrees) are related by the formula: Example 1: Draw the graph of y = sin2θ 2π
2π
or b =
b
Period
360
360
Period =
or b =
b
Period Period = (0 ≤ θ ≤ 2π ) The first step in graphing this trig function is to find the period. 2π
b
2π
Period =
2
Period = π Note that
tanθ graphs
do not use
these
equations for
the period
and bvalue. Period = Once we know the period, draw the graph
from 0 to 2π, since that is the specified domain. Example 2: Draw the graph of y = cos 1
θ
2 (0 ≤ θ ≤ 4π ) The first step in graphing this trig function is to find the period. 2π
b
2π
Period =
0.5
Period = 4π
Period = Once we know the period, draw the graph
from 0 to 4π since that is the specified domain. Pre – Calculus Math 40S: Explained!
www.math40s.com 46 TRIGONOMETRY LESSON FIVE
PART II  THE B VALUE
Given a graph, you must find the b value before you can write the equation.
Example 1: Find the cosine
equation of the following graph: Step 1: First you need Step 2: Once you identify the Step 3: Now that to draw a rectangle
around the cosine
pattern. In this graph,
we can easily see a
cosine patten going period, find b by performing
the following calculation: we have the b value,
and a cosine pattern
is identified, we can
write the equation : from 0 to 2π
3 2π
Period
2π
b=
2π
3
3
b = 2π ×
2π
b=3
b= y = cos3θ Example 2: Find the sine
equation of the following graph:
Step 1: First you
need to draw a
rectangle around the
sine pattern you
want to use. In this
graph, we can easily
see a sine pattern
going from 0 to 10π Step 2: Once you
identify the period, find
b by performing the
following calculation: 2π
Period
2π
b=
10 π
1
b=
5
b= Step 3: Now that
we have the b value,
and we identified a
sine pattern, we can
write the equation: y = sin 1 θ
5 Pre – Calculus Math 40S: Explained!
www.math40s.com 47 TRIGONOMETRY LESSON FIVE
PART II  THE B VALUE
Questions: For each of the following graphs, write the equation:
1) 2) 3) 4) For each of the following equations, draw the graph:
5) 7) y = sin2θ y = cos3θ 6) 1
y = cos θ
3 8) 1
y = sin θ
4 Pre – Calculus Math 40S: Explained!
www.math40s.com 48 TRIGONOMETRY LESSON FIVE
PART II  THE B VALUE
Answers:
2
5 1) cos4θ b= 2) cos θ 2π 2π
2
=
= 2π × = 4
π
P
π
2 1
2 3) sin θ b= b= 2π 2 π 2
=
=
P 5π 5 4) sin6θ 2π 2 π 1
=
=
P 4π 2 b= 2π 2π
3
=
= 2π × = 6
π
P
π
3 2π
b
2π
P=
1
3
P= 2π
b
2π
P=
2
P =π
P= 5) 6) P = 2π ×
P = 6π 3
1 2π
b
2π
P=
1
4
P= 2π
b
2π
P=
3 P= 7) 8) P = 2π ×
P = 8π 4
1 Pre – Calculus Math 40S: Explained!
www.math40s.com 49 TRIGONOMETRY LESSON FIVE
PART III  THE C VALUE
The phase shift is the horizontal translation applied to a trig graph. It is the number
added or subtracted to θ inside the equation.
Phase shift is represented by the letter “c” in y = sin(θ ± c)
Notice in the following graphs that you will do the opposite of what the sign is.
The + will move the graph left, and the — will move the graph right. y = sinθ
y = sin(θ  π )
The π means we
move the graph
right by π units. y = sinθ π⎞
⎛
y = sin ⎜ θ + ⎟
2⎠
⎝ π means we
2
move the graph left
The + by π 2 units. Not all graphs are going to be given as one cycle, since trig graphs can go forever in both
directions! A phase shift will shift everything horizontally by the same amount, so it’s still
easy to graph.
y = cosθ π⎞
⎛
y = cos ⎜ θ + ⎟
2⎠
⎝ The + π means we
2
move the graph left by π 2 units. Pre – Calculus Math 40S: Explained!
www.math40s.com 50 TRIGONOMETRY LESSON FIVE
PART III  THE C VALUE
Quite often, a graph will be given with ticks where no radian measure is indicated. In these
questions, we need to figure out what the exact value of each tick is first.
In this graph,
we see six
ticks between
0 and π, so
each tick must zooming in from 0 to π be 30º or In this graph,
we see four
ticks between
0 and π, Each
tick must be zooming in from 0 to π π 45º, or π 6 4 It is always possible to write at least one sine equation and one cosine equation for the
same trig graph.
Notice how ticks are given in the graph between π and 0.
Think in terms of degrees for a moment. If we have 180º
and 6 ticks, that makes each one 30º. So, if our sine pattern
starts at the fourth tick back, that would be 120º, or
in radians,  2π ⎛
⎝ . The sine equation is y = sin ⎜ θ + 2π ⎞
⎟.
⎠ 3
3
Likewise, we can see that if we were thinking in terms of
cosine, the cosine pattern starts one tick back, at 30 º ⎛
⎝ The cosine equation would be y = cos ⎜ θ + 2π ⎞
⎛
y = sin ⎜ θ +
⎟
3⎠
⎝ ⎞
⎟
6⎠ π π⎞
⎛
y = cos ⎜ θ + ⎟
6⎠
⎝ OR
Always try to find an
“upright” trig pattern to
derive the equation. Pre – Calculus Math 40S: Explained!
www.math40s.com 51 TRIGONOMETRY LESSON FIVE
PART III  THE C VALUE
Questions: For 1 & 2, write the sine equation. For 3 & 4, write the cosine equation.
1. 2. 3. 4. For 5 & 6, draw the sine graph. For 7 & 8, draw the cosine graph.
π
5. y = sin(θ + )
6.
3 7. y = cos(θ + π
4 ) 8. y = sin(θ  y = cos(θ  π
4 5π
6 ) ) Pre – Calculus Math 40S: Explained!
www.math40s.com 52 TRIGONOMETRY LESSON FIVE
PART III  THE C VALUE
Answers: 1.
sin(θ  π
4 We can draw a rectangle around the sine pattern closest to
the origin. There are four ticks between 0 and π, so each one
is 45º. Since the sine pattern starts on the first tick to the right, ) the equation is y = sin(θ  2. 2π
sin(θ )
3 π
4 ) We can draw a rectangle around the sine pattern closest to the
origin. There are six ticks between 0 and π, so each one is 30º.
Since the sine pattern starts on the fourth tick to the right,
which is 120º, the equation is y = sin(θ  3. cos(θ + 4. cos(θ + π
2 4 3 ) We can draw a rectangle around the cosine pattern closest to
the origin. There are six ticks between 0 and π, so each one is
30º. Since the cosine pattern starts on the third tick to the left, ) π 2π which is 90º, the equation is y = cos(θ + ) π
2 ) We can draw a rectangle around the cosine pattern closest to
the origin. There are four ticks between 0 and π, so each one
is 45º. Since the cosine pattern starts on the first tick to the
left, which is 45º, the equation is y = cos(θ + 5. 4 ) 6. 7. π 8. Pre – Calculus Math 40S: Explained!
www.math40s.com 53 TRIGONOMETRY LESSON FIVE
PART IV  GRAPHING B AND C
We will now look at trig graphs with the form: y = sinb(θ + c)
”b” is used to find the period using the formula: Period = 2π
b “c” is the letter used to represent phase shift.
When combining b & c, we should follow a particular order.
First apply the period, then the phase shift.
π Example 1: Graph y = sin2(θ  ):
3 y = sinθ y = sin2(θ  y = sin2θ
(Remember to use your formula to
find the period, which is π) 1
2 Example 2: Graph y = cos (θ +
y = cosθ π
4 y = cos π ) 3 (Move the graph two ticks right, since
each tick is 30º and we want 60º) ): 1
2 θ (Remember to use the formula to find
period, which is 4π) y = cos 1
2 (θ + π
4 ) (Now move the graph one tick left,
since each tick is 45º) Sometimes, the bvalue is attached to θ inside the brackets. In the equation y = sin(2θ – π
3 ), we MUST factor out the 2 before graphing. The reason for doing this is that we can now easily read off the phase shift. y = sin(2θ y = sin2(θ  π
3 π 6 )
) When you pull out
the 2, divide each
term in the original
brackets by 2. Pre – Calculus Math 40S: Explained!
www.math40s.com 54 TRIGONOMETRY LESSON FIVE
PART IV  GRAPHING B AND C
Questions: Graph the following equations:
1) y = sin 2 (θ  π )
3
2 1
3 3) y = cos (x  π ) π 5) y = sin(2θ  )
3 π 2) y = sin2(θ  )
4 4) 6) y = cos(2θ  π ) y = cos(4θ + π ) Pre – Calculus Math 40S: Explained!
www.math40s.com 55 TRIGONOMETRY LESSON FIVE
PART IV  GRAPHING B AND C
Now use the formula 1) 3) 5) First graph
y = sinθ First graph
y = cosθ First graph
y = sinθ b
to find the period, which is 3π Move the graph three
ticks right, since each
tick is 30º and we want a
shift of 90º The period
is 6π Move the graph one tick
right to π. The period is π
(Don’t forget to
factor out the 2) Move the graph right one
unit to 30º, or π/6 P= 2π 2) 4) 6) First graph
y = sinθ First graph
y = cosθ First graph
y = cosθ Now use your formula to
find the period, which is π. The period is π
(Don’t forget to
factor out the 2) The period is π/2
(Don’t forget to
factor out the 4) Finally move the graph one
tick to the right, since each
tick is 45º and we want a
shift of 45º Move the graph right two
units to 90º, or π/2 Move the graph left one unit to
45º, or π/4 Pre – Calculus Math 40S: Explained!
www.math40s.com 56 ...
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 Winter '10
 KISCABEAN

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