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Graphing (Method for sin\cos, cos example given)
Graphing (Method for cot)
Graphing (sec\csc, use previous cosgraph from example above)
Tangent Graphing, blast from the past
Composite functions f(f -1(x))
Graphing (Method for tan)
Write sin\cos equation given graph
Basic Sum\Difference formula usages.(Using cos)
Basic Half Angle formula usages.
Equation (Not so basic)
Identities (2)
14
Equation (Basic)
Identities (1)
First timers, the buttons above take you to a topic. The home button brings you back. If you find a mistake, IM me at kimtroymath or e-mail me at [email protected]
There are MANY things on this test, it’s a big one. This powerpoint does NOT cover everything. Wait for the review sheet before the test for info, and use this powerpoint to help you cover some of the materials. MATERIAL FOR CH 6 TEST IN RED
A) Factor out the coefficient of x, and use even-odd properties to simplify
1) Find Amplitude and period
2) Find Phase Shift, and vertical shift
3) Find starting and ending x-coordinates
4) Divide into 4 equal parts
5) Label key points
6) Connect
2)3cos(3 xy
2))3(cos(3 xy
Remember, cos(x) = cos(-x)
2))3(cos(3 xy
You will always do this, this is part of your ‘work’ on a test and is required
Amplitude =
T =
P.S. =
V.S. =
3
3
2
2
3 2
Starting point is phase shift. Ending point is Phase shift + Period
3 5
+
You will take the starting and ending points and find the average, then find the average again to break it up into four equal regions
42
53
4
2
7
2
43
2
7
2
9
2
54
2
9
You want to study the sine and cosine graphs. Remember:
Sine 0, 1, 0, -1, 0
Cosine 1, 0, -1, 0, 1
You are basically performing transformations on those key points
1
A) Factor out the coefficient of x, and use even-odd properties to simplify
1) Find Vertical Stretch and period
2) Find Phase Shift, and vertical shift
3) Find starting and ending x-coordinates
4) Divide into 4 equal parts
5) Label key points
6) Connect
1)2cot(2 xy
4
1
12
2cot2
xy 2 1
2
Vertical = Stretch
Period =
Phase Shift =
Vertical Shift =
||
2
2
START
START
2
+END
22
END
4
3
22
4
3
Find the average, then find the averages again.
8
5
24
32
8
7
24
3
8
58
7
cot, asymp 1 0 -1 asymp
A) Factor out the coefficient of x, and use even-odd properties to simplify
1) Find Vertical Stretch and period
2) Find Phase Shift, and vertical shift
3) Shift zero (the middle)
4) Divide the period in half, add and subtract from the middle, sketch asymptotes.
5) Perform transformations.
6) Connect
142
tan2
xy
12
1
2tan2
xy
2
2
11
Vertical = Stretch
T =
P.S. =
V.S. =
|| 2
Remember,
it’s π\22
2
2
1
1
12
2
2
T11
2
1
2
1
2
3
2
1
-1, 1 between
Go between the asymptotes and the middle and put -1, 1, then transform.
A) First, sketch the cos graph, stating all the information as your for cosine.
1) At the ‘zeros’ of cosine (or the middle), sketch asymptotes.
2) At the maxes and mins (tops and bottoms), make your U’s
2)3cos(3 xy
3 51
2)3sec(3 xy
You aren’t going to need three cycles. Probably just one cycle.
2
1
Write the equation of the sin and cos graph.
Remember, for sin and cos, the amplitude, period, and vertical shift are all the same, only the phase shift is different.
bxAy
bxAy
))(cos(
))(sin(
Amplitude. You can use common sense, how far is the max from the middle, or you can use the formula.
Vertical Shift, how much did the middle move from the x-axis? Or you can use the formula.
Clear
Vertical Shift Amplitude
Clear
Period
Clear
cos Phase Shift
Clear
sin Phase Shift
Clear
Concept, V.S. is average of max and min. So you add and divide by 2.
Concept, Amplitude is the distance between max and min, so you subtract (distance) then divide by 2.
2
minmax..
bSV
22
)4(0
b
2
minmaxAAmp
22
)4(0
Amp
Max
Min
2
2
2
2
The period of sin and cos are the same, so it’s easiest (IMHO) to find the period using cosine. To do that, find out how far apart the maxes are.
Max4
Max4
5
44
5TPeriod
T
2
2
2
2
Cosine starts at the top (1 0 -1 0 1). So find a max, the x-coordinate is a possible phase shift.
There are many possible solutions. For this problem, you could use -3pi\4, pi\4, 5pi\4, etc.
4
Sin starts in the middle, then goes up (0 1 0 -1 0). Find a point in the middle where the graph goes up afterwards. That is one possible phase shift.
-pi, 0, and pi are all viable options.
middle
Then up
middle
Then up
middle
Then up
0
You could try to graph tangent using old style transformations if you wanted to.
Factor out the coefficient of x.
Horizontal
Stretch:
Reflect:
Shift:
Vertical
Stretch:
Reflect:
Shift:
Asymptote changes are only affected by horizontal transformations.
142
tan2
xy
12
1
2tan2
xy
2
None
2
1Left
2
None
1Up
2
1,4
0,01,42
xx
11,2
10,01,
2
11
xx
2
1
1
2
11,00,
2
11,1
2
3
xx
2,00,2
12,1
3,01,2
11,1
Key Points
I recommend the other method for faster graphing.
2
2
This is different than the other tangent graph because there is an addition sign inside, not a subtraction sign.
Composite trig functions.
1) Set up triangle in the correct quadrant.
1) Pythagorean Theorem may be necessary. (r always positive)
2) Find solution using correct sides.
20
0:sec22
:csc
0:cot22
:tan
0:cos22
:sin
,,
xx
xx
xx
xx
IIIQuadrantsIVIQuadrants
3
7secsin 1
7
3
x
rsec
2
r
ysin
7
2
4
3tancos 1
Note, secant negative means it’s in quadrant II.
x
ytan
34
r
xcos
55
4
5
13csccos 1
y
rcsc
135
Remember, r is positive.
12
r
xcos
13
12
Basic sum\difference formula usages.
Note, only showing examples for sin. These problems may show up in cos and tan format. If you need to do cot, sec, or csc, use the formula of their reciprocal counterparts, then take the reciprocal of the solution. You need to be careful though, you will have to rationalize the denominator at times. The CONJUGATE will be very helpful.
In this example, to find csc, I use sin to find a solution, and since csc is the reciprocal of sin, I take the reciprocal of my answer. The work is shown on how to break it down. Watch for the conjugate.
BABABA
BABABA
sincoscossin)sin(
sincoscossin)sin(
Application
Clear
Reverse
Clear
Note
Clear
12
5sin
Break it up into common radian values. A chart may be helpful. (Note, use same denominators)
12
12
12
6
2
12
10
6
5
12
4
3
12
9
4
3
12
3
4
12
8
3
2
12
2
6
Many combinations are possible. These add up to equal 5pi over 12.
46sin
Apply appropriate formula
sincoscossinsin
4
4
4
6
6
6
2
1
2
2
2
3
2
2
4
62
50sin110cos50cos110sin
BABABA sincoscossin)sin(
110 11050 50 )sin(
)60sin(
2
3
Match up the expression with the correct formula.
These problems are designed to give you a familiar unit circle value.
62
4
)62(4
62
62
62
4
462
1
125
sin
1
12
5csc
Example
2
cos1 A
13
12
Basic Half-Angle formula Usage
A
AA
A
A
AA
AA
AA
cos1
sinsin
cos1
cos1
cos1
2tan
2
cos1
2cos
2
cos1
2sin
QIIIinisAA ;13
5sin
What quadrant is A\2 in?
2
Acos Find
Set up triangle, use original A when plugging into formula.
Your solution will be plus OR minus, not plus and minus. Use the quadrant of A\2 to determine SIGN!
2
3 A
222
4
3
22
AQII
Cosine is negative in quadrant II, so we will use the negative sign.
513
12
13
12cos
A
A2
131
26
26
26
1
15sin These may also be done with sum\difference formulas (45o – 30o), but see if doubling it may give you a common unit circle measure.
2
30sin
2
cosA1
Look at 15o, it’s in quadrant I, so sin is POSITIVE in quadrant I.30
30
2
30
2
ASo
A
223
1
2
32
4
32
22
32
Common denominator
These problems will also involve radians. They work in a similar fashion.
Equations (Basic)
cottan,
2seccsc,cos,sin,
2
1
23
2cos
Give the general formula for all the solutions.
?2
1cos(x) does Where
23
2 23
2
3
3
5k2 kk;2 2every repeats cos
23
2 xlikeit ofcan think You
23
2but really x,not sIt'
x
k 263
2
k 34
k 26
7
3
2
k 34
7
Be careful, sometimes you can combine general formulas.
)[0,2between for solutions all Find
integer.an is
k remember, k,for in values Plug
k
1k0k-1k
4
134
54
4
74
114
19
4
7
)[0,2
betweennot ones out the Cross
This does NOT cover all possible types of equations. Some common things to watch out for regarding equations:
1) Move everything to one side.
2) Use properties when possible (Sum to Product, pythagorean, double angle)
3) Many times, changing things into the same trig function may be helpful.
4) Factoring may occur many times.
5) Remember to plus\minus when square rooting both sides.
6) You can combine general formulas sometimes.
012cossin AA01)sin1(sin 2 AA
01sin1sin 2 AA
0sinsin 2 AA0)sin1(sin AA
Give the general formula for all the solutions.
1sin
0sin10sin
A
AA
kkA
kAkA
2
22
32
k
kAkA 22
3
Identities: Pg 513: 45 RULE: WORK ON ONE SIDE ONLY!
Remember the question mark.
Helpful Items, in no particular order.
1) Changing to sin\cos helps.
2) Look for Pythagorean, double angle, product-to-sum, reciprocal, even\odd identities.
3) Look at the other side.
4) Conjugates and multilying by one helps.
5) Combining or splitting up fractions is also helpful.
6) Factoring may be helpful.
7) Work on more complicated side.
1cot2cotcot2 2?
AAA
1cot2tan
1cot2 2
?
AA
A
1cot
tan1tan21
cot2 2?
2
A
AA
A
1cottan2
tan1cot2 2
?2
A
A
AA
1cot)tan1(cot 2?
22 AAA
1cottancotcot 2?
222 AAAA
1cot1cot 22 AA
I noticed there was ‘cot’ on the other side, that’s why I didn’t change to sin\cos in this case.
Identities: Pg 512: 30 RULE: WORK ON ONE SIDE ONLY!
Remember the question mark.
Helpful Items, in no particular order.
1) Changing to sin\cos helps.
2) Look for Pythagorean, double angle, product-to-sum, reciprocal, even\odd identities.
3) Look at the other side.
4) Conjugates and multilying by one helps.
5) Combining or splitting up fractions is also helpful.
6) Factoring may be helpful.
7) Work on more complicated side.
AA
Asin
sin1
cos1
?2
AA
Asin
sin1
sin11
?2
AA
AAsin
sin1
)sin1)(sin1(1
?
AA sinsin11?
AA sinsin
AA
AAsin
sin1
cossin1 ?2
AA
AAsin
sin1
)sin1(sin1 ?2
AA
AAsin
sin1
sinsin ?2
AA
AAsin
sin1
)sin1(sin ?
AA sinsin
Here is another method
14