Download Graphing Trigonometric Functions: Day 1 Pre

Graphing Trigonometric Functions: Day 1
Pre-Calculus
Learning Targets
1. Graph the six parent trigonometric functions.
2. Apply scale changes to the six parent trigonometric functions.
Complete the worksheet Exploration: Introduction to Ch. 4B BEFORE you watch the video for day 1.
From the exploration, notice we are graphing the ordered pairs ( ,sin  ) and ( , cos  ) which gives us the graph of the
sine function and the cosine function. You may recall these parent functions from first semester.
Sine Function: f ( x )  sin x
Cosine Function: f ( x )  cos x
Some new terminology…

Periodic Functions:

Period:

Amplitude:
Example 1: List the amplitude and period. Then graph at least one period of the function. Be sure to label the scale of
each axis.
a) y  2 sin x
 x
2
c) y   cos  
b) y  cos  2 x 
d) y 
4b–1
1  x
sin  
3 4
Example 2: Write two equations of the cosine function whose amplitude is 5 and period is

3
.
Example 3: Write the equation for the given graph.
1
What about Tangent?
Tangent Function: f ( x )  tan x

Period:

Amplitude:

Vertical Asymptotes:
x
2
Example 4: Graph at least one period of y  tan   . Be sure to label your axes and clearly identify the asymptotes.
We will discuss the graphs of the reciprocal functions together in class!
4b–2
Graphing Trigonometric Functions: Day 2
Pre-Calculus
Learning Targets
1. Apply translations to the six parent trigonometric functions.
2. Given the attributes of a trigonometric function write the equation of a trig. function.
3. Given a graph, identify the attributes of the function and write its equation.
To review our translations…A number added inside effects the horizontal direction, but it’s backwards, and a number
added outside effects the vertical direction. When we apply these transformations to the trig functions, we get…
y  a sin b( x  c )   d

Amplitude:

Period:

Phase Shift:

Vertical Shift:
or
y  a cos b( x  c )   d
Example 1: List the amplitude, period, phase shift and vertical shift. Then graph at least one period of the function. Be
sure to label the scale of each axis.


b) y  cos  x 
a) y  4 sin x  2
 
 
c) y  sin 3  x 
 

 1
2
x
d) y  tan    5
4

3  
4b–3
For this last example, be careful…the phase shift is NOT


e) y  3sin  2 x 

2
!

 1
2
Example 2: Write two equations of the cosine function whose amplitude is 2, period is 6 , phase shift is
and vertical shift is –5.
Example 3: Write the equation for the given graph using…
a) the Sine Function
b) the Cosine Function.
4b–4
2
3
4.7 Inverse Trigonometric Functions
Pre-Calculus
Learning Targets
1. Use the appropriate notation for inverse trigonometric functions.
2. Graph the inverse Sine, Cosine and Tangent functions.
3. List the correct Domain and Range of the inverse functions.
4. Find an exact solution to an expression involving an inverse sine, cosine or tangent.
5. Find the composition of trig functions and their inverses.
Inverse Sine
Inverse Cosine
Inverse Tangent
y  sin 1 x
y  arcsin x
y  cos 1 x
y  arccos x
y  tan 1 x
y  arctan x
D: [–1, 1]
D: [–1, 1]
D: (–∞, ∞)
R:
  ,  
 2 2 
R:
  ,  

 2 2
 0,  
R: 
Example 1: Find the exact value (in radians).
a) cos-1 0
b) sin-1 0
1
e) cos 1  
2
f) tan 1  3

 1 

 2
c) arcsin 
 3

 2 

g) arcsin 
4b–5
d) arctan 1
 3

 2 
h) arccos 
Compositions with Inverse Functions
 Work from the inside out.
 Remember domain and range restrictions.
Example 2: Evaluate each expression.


a) sin arctan  3 

  5
b) cos 1  cos 
  3



Example 3: Find the algebraic expression equivalent to the given expression.
1
a) sin (cos x)
1
b) cot (sin 2 x)
4b–6
Solving Prob
blems with Trigonometry
4.8 S
Prre-Calculus
Learnning Targets
11. Set up and
d solve appliccation problem
ms involving right trianglee trigonometryy.
22. Use overlapping right triangles
t
to so
olve word pro
oblems includding the use oof indirect measurement.
33. Solve problems involving simple haarmonic motio
on.
Next we will applyy what we kno
ow about the trigonometricc functions, aand their inverrses, to solve real world appplication
problems. The mo
ost important part of these types of quesstions is an acccurate, detaiiled picture.
Anglle of Elevation: The acute angle measurred from a ho
orizontal
line U
UP to an objeect.
Anglle of Depressiion: The acutte angle meassured from a horizontal
h
line D
DOWN to an object.
Angle of Deprression
Angle off Elevation
Exam
mple 1: From
m a boat on thee lake, the ang
gle of elevatio
on to the top of a cliff is 2042. If the base of the clliff is 1394
feet ffrom the boatt, how high is the cliff (to the
t nearest foo
ot)?
Exam
mple 2: The bearings
b
of tw
wo points on the
t shore from
m a boat are 1 15 and 123. Assume the two points aare 855 feet
apartt. How far is the boat from
m the nearest point
p
on shoree if the shore is straight annd runs north--south?
4b–7