Unformatted text preview: Pre – Calculus Math 40S: Explained!
www.math40s.com 82 Trigonometry Lesson 9
Part I  Graphing Other Trig Functions
Graphing y = tanθ & y = cotθ y = tanθ The vertical lines you see are called asymptotes.
They are places where the graph is undefined.
Recall that tanθ = sinθ
cosθ and cotθ = cosθ
sinθ tanθ is undefined at the angles where cosθ is
equal to zero. ⎡ π 3π ⎤
⎢2, 2 ⎥
⎣
⎦ Likewise, cotθ is undefined whenever
sinθ is equal to zero. [0, π ] For tanθ, we can see from the graph that It is important you state the general solution
of the asymptotes for each graph. the first positive asymptote occurs at avalue: We only use the term
amplitude in describing the
graphs of sinθ and cosθ.
The other four trig graphs are
not “closed in”, they go up &
down forever.
So, we simply call the avalue
the vertical stretch. 2
All asymptotes are exactly π units away
from each other. π b and b = . The general equation of the asymptotes
is: x= π 2 ± nπ
y = cotθ bvalue & period: IMPORTANT!
The period of a basic tanθ or
cotθ graph is π, not 2π like sinθ
and cosθ. Thus, we have the
following formulas: Period = π π Period cvalue: No difference from
sinθ and cosθ, but remember
to move your asymptotes if
you shift the graph.
dvalue: No difference from
sinθ and cosθ. We always write the general solution of
tanθ & cotθ asymptotes in the following way: For cotθ, we can see from the graph that
the first positive asymptote occurs at 0,
and all asymptotes are exactly π units
away from each other.
The general equation of the asymptotes
is:
x = 0 ± nπ , or simply, x = ± nπ x = Angle of first positive asymptote ± n(Period) Pre – Calculus Math 40S: Explained!
www.math40s.com 83 Trigonometry Lesson 9
Part I  Graphing Other Trig Functions
Graphing y = cscθ & y = secθ y = cscθ
Grey = sinθ As with the previous graphs, the vertical lines you
see are asymptotes. and they occur where
the graph is undefined.
Recall that csc θ = 1
sin θ and secθ = 1
cosθ At the angles where cosθ is equal to zero,
the graph is undefined
Likewise, cotθ is undefined whenever
sinθ is equal to zero. For cscθ, we can see from the graph
that the first positive asymptote occurs
at 0, and all asymptotes are exactly π
units away from each other.
The general equation of the
asymptotes is: avalue: Vertical stretch, makes
graphs taller and more narrow.
bvalue & period: For the
reciprocal graphs cscθ and secθ,
they have a period of 2π, so we can
use the formulas we’re used to: Period = 2π
b and b = x = 0 ± nπ , or simply, x = nπ 2π
Period y = secθ cvalue: No difference from sinθ
and cosθ, but remember to move
your asymptotes if you shift the
graph. Grey = cosθ dvalue: No difference from sinθ
and cosθ. For secθ, we can see from the graph that
the first positive asymptote occurs at π 2
All asymptotes are exactly π units away
from each other. . The general equation of the asymptotes is: x= We always write the general solution of
cscθ & secθ asymptotes in the following way: π 2 ± nπ ⎛ Period ⎞
⎟
⎝2⎠ x = Angle of first positive asymptote ± n ⎜ Pre – Calculus Math 40S: Explained!
www.math40s.com 84 Trigonometry Lesson 9
Part I  Graphing Other Trig Functions
Find the equation of each of the following graphs. Also, state the general solution of the asymptotes. 1. 2. 3. 4. Draw the graph for each of the following equations, and state the general solution of the asymptotes.
5. y = cot3θ 1
7. y = tan θ
2 1
6. y = sec θ
3 8. y = csc2θ Pre – Calculus Math 40S: Explained!
www.math40s.com 85 Trigonometry Lesson 9
Part I  Graphing Other Trig Functions
1) The period of this secθ graph is π.
(The length of one complete cycle must contain both the top U and the upsidedown U.) 2π Now find the bvalue: b = = Period General Solution: x = π ±n 4 2π π Equation is: y = sec2θ. = 2. π
2 2) The period of this tanθ graph is π/3. We can tell since each tick is 30º and one cycle uses two ticks.
π Now find the bvalue: b = = Period π
π =π× 3 π = 3. Equation is y = tan3θ . 3 General Solution: x = π
6 ⎛π ⎞
⎟
⎝3⎠ ± n⎜ π 3) The period of the cotθ graph is 2π. Now find the bvalue: b =
General Solution: Period x = ± n ( 2π ) 2π 4) The period of the cscθ graph is 6π. Now find the bvalue: b =
General Solution: Period x = ± n ( 3π ) 5) = = π
2π 2π
6π 1 = 1 = 1
y = csc θ
3 3
2 1 1 −π π −6 π−5 π−4 π−3 π−2 π −π 1 π 2 π 3 π 4π 5 π 6π 1 General
Solution:
⎛π ⎞
x =±n⎜ ⎟
⎝ 3⎠ 2
3 2
3 8) 3 General
Solution:
3π
± n( 3π )
x=
2 3 2 2 1 −4 π −3 π −2 π Equation is 3 2 7) 1
y = cot θ
2 2 6) 3 Equation is 1 −π π 2π 3π 4π −π 1
2
3 π 1 General
Solution:
x =π ± n( 2π ) 2
3 General
Solution:
nπ
x =±
2 Pre – Calculus Math 40S: Explained!
www.math40s.com 86 Trigonometry Lesson 9
Part II  Summary of Trig Functions
1
Example 1: Given the equation: y = csc θ , find the following:
2
Questions:
a) avalue
b) bvalue Answers:
a) 1
b) ½ c) Period c) Period = d) Phase shift
e) Vertical Displacement
f) Domain
g) Range
h) xintercepts
i) yintercepts
j) Equation of asymptotes d) None
e) None
f) x ≠ ± n(2π )
g) y ≤ − 1 , y ≥ 1
h) None
i) None
j) x = ± n(2π ) 3 2π 2π
=
= 4π
b 0.5 Example 2: Given the equation: h(t) = 7sin
Questions:
a) avalue
b) bvalue
c) Period
d) Phase shift
e) Vertical Displacement
f) Domain
g) Range
h) xintercepts
i) yintercepts
j) Equation of asymptotes Answers:
a) 7
2π
b)
5
c) 5
d) 1.25 right
e)15 up
f) t ∈ R
g) 8 ≤ h(t ) ≤ 22
h) None
i) 8
j) None 2
1 −4 π −3π −2 π −π π 2π 3π 4π 1
2
3 2π
(t 1.25)+15 , find the following:
5
2
5
2
0
1
5
1
0
5 2 4 6 8 1
0 Reminder: To find xintercepts, use
2nd
Trace
Zero.
You should always state general
solutions for xintercepts.
(unless the domain is specified)
Find yintercepts using:
2nd
Trace
Value
x=0 Pre – Calculus Math 40S: Explained!
www.math40s.com 87 1 www.math40s.com 10 9. 8. 7. 6. 5. 4. 3. 2. 1. Equation avalue bvalue Period Phase
Shift Vertical
Displacement Domain Range xintercepts yintercepts Equation
Of
Asymptote Trigonometry Lesson 9 Part II  Summary of Trig Functions Pre – Calculus Math 40S: Explained! 88 Trigonometry Lesson 9
Part II  Summary of Trig Functions
avalue 1. bvalue 2 4 Period Phase
Shift π π 2 4 right Vertical
Displacement Domain Range xint yint Equation
Of
Asymptotes 5 up θ ∈R 3≤ y ≤7 None 5 None d ∈R −15.5 ≤ T ≤ 26.5 21 2π
365 365 118 right 5.5 up 3. 1 1
2 2π None None 4. 1
2 2 π 45º right 1 up θ ∈R 5. 18.5 2π
365 365 28 right 4.5 up t∈R 6. 1
2 1
3 6π 3 up θ ∈R 7. 1 1
3 6π None None 2. π
2 left x ≠ π ± n ( 2π ) x≠ 3π
± n (3π )
2 x ≠ ± n(2π ) 316 ± n(360 ) 13.31 None x = π ± n ( 2π ) y∈R ± n(2π ) 0 0 .5 ≤ y ≤ 1.5 None 1 None 303 ± n(360 ) 20.9 None None 3.25 None None 1 y∈R π ± n(2π ) None x = ± n ( 2π ) None 0.29 None 19.6 None −14 ≤ h ≤ 23 2.5 ≤ y ≤ 3.5 134 ± n(360 ) y ≤ −1
&
y ≥1 8. 1 1
2 2π None None 9. 2 1
3 6π π right 2 up θ ∈R 0≤ y≤4 20.1 2π
300 300 265 right 6.2 up t∈R −13.9 ≤ y ≤ 26.3 10 102 ± n(360 ) x= 3π ± n (3π ) 2 130 ± n(360 )
250 ± n(360 ) Pre – Calculus Math 40S: Explained!
www.math40s.com 89 ...
View
Full
Document
This note was uploaded on 02/09/2011 for the course 168 comm 168 taught by Professor Kiscabean during the Winter '10 term at UCLA.
 Winter '10
 KISCABEAN

Click to edit the document details