6.1 Sinusoidal Graphs
Periodic Function:
A function f is periodic if its values repeat at regular intervals.
Graphically, this means that if the graph of f is shifted horizontally by p units, the new graph is identical to the
original.
In Function Notation, periodic means that, for all x in the domain of f,
π(π₯ + π) = π(π₯) The smallest positive constant p for which this relationship holds for all values of x is called the period of f.
The midline of a periodic function is the horizontal line midway between the functionβs maximum and minimum
values.
The amplitude is the vertical distance between the functionβs maximum (or minimum) value and the midline.
Graphs of the Sine and Cosine Functions (Note: Sinusoids are considered to be the general form of the sine and
cosine function)
π‘ 0 π
4
π
2
3π
4 π
5π
4
3π
2
7π
4 2π
sin π‘
cos π‘
Period: _________
Midline: ________
Amplitude: ______
Period: _________
Midline: ________
Amplitude: ______
Amplitude
π¦ = π sin π‘,
then |π| is the amplitude of the graph.
Ex. π¦ = sin π‘
π¦ = 2 sin π‘
π¦ = β sin π‘
Midline
The graph of π¦ = π(π‘) + π is the graph of π¦ = π(π‘) shifted vertically by d units.
Note: If π¦ = sin π‘ + π and π¦ = cos π‘ + π, we have midlines π¦ = π.
Ex. π¦ = cos π‘ + 2
Note:
Midline of both graphs is π¦ = 0.
Amplitude is 1.
Note: If a is negative, the graph is reflected across the t-axis
Shifted Sine and Cosine Curves Given an equation in the form khtBAtf )(sin)( or khtBAtf )(cos)(
amplitude = |π΄|
period, B
P2
midline π¦ = π
h is the horizontal shift of the function
An appropriate interval on which to graph one complete period is[β, β + (2ππ΅β )]
Note: horizontal shift means: the curves shifted horizontally by an amount |β|. They are shifted to the right if h >
0 or to the left if h < 0.
Period
Ex. π¦ = sin(2π‘) Ex. π¦ = sin(π‘2β )
Horizontal Shift
Ex. π¦ = 2sin(π‘ + π2β ) Ex. π¦ = β3sin(2π‘ β π
2β )
For the equation below, determine the amplitude, midline, period, and horizontal shift (indicated the direction).
π¦ = β5sin(ππ‘ + 2) β 4
For the graphs below, determine the amplitude, midline, period, and horizontal shift (indicated the direction), then write an
equation for the graph.
5. 7.
17.
23. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o'clock
position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The
function ( )h t gives your height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h t
b. Find an equation for the height function h t
c. How high are you off the ground after 5 minutes?
6.2 Graphs of the Other Trigonometric Functions
On a unit circle, we know that
π₯ = and π¦ =
Find the slope of the line ππ ,
Graph of the Tangent Function (π = ππππ½)
Period =
Range:
Domain:
Period =
Range:
Domain:
P(x, y)
1
ΞΈ y
x 0
Note: The functions π¦ = π΄ tan(π΅π₯) and π¦ = π΄ cot(π΅π₯), where π΅ > 0, have period π
π΅.
Graph of π¦ = csc π₯ Graph of π¦ = sec π₯
Find the period and horizontal shift of each of the following functions,
5. 2tan 4 32f x x
7. 2sec 14
h x x
and then graph the function
21. If tan 1.5x , find tan x
23. If sec 2x , find sec x
Period =
Range:
Domain:
Period =
Range:
Domain:
Note: The functions π¦ = π΄ csc(π΅π₯) and π¦ = π΄ sec(π΅π₯), where π΅ > 0, have period 2π
π΅.
6.3 Inverse Trigonometric Functions
Recall:
If f is one-to-one function with domain A and range B, then its inverse f-1
is the function with domain B and range
A defined by
πβ1(π₯) = π¦ β π(π¦) = π₯ Therefore,
f-1
is the rule that reverse the action of f. (From output back to input)
Note: πβ1 reads f inverse or inverse of f, and xf
xf11
If a function has an inverse, it is said to be invertible.
The Inverse Sine Functions:
The inverse Sine function is the function π ππβ1 with domain [-1, 1] and range
2,
2
defined by
sinβ1 π₯ = π¦ β sin π¦ = π₯
The inverse Sine function is also called the arcsine function, denoted by xarcsin .
Evaluate the following expressions
1. 1 2sin
2
2. 1 1sin
2
3.
1sin 1
4.
6sinsin 1
5.
2
1sinsin 1 6. 5sinsin 1
Domain:
2,
2
Domain: [-1, 1]
Range: [-1, 1] Range:
2,
2
The Inverse Cosine Functions: The inverse Cosine function is the function πππ β1 with domain [-1, 1] and range ,0 defined by
cosβ1 π₯ = π¦ β cos π¦ = π₯
The inverse Cosine function is also called the arccosine function, denoted by xarccos .
Evaluate the following expressions
1. 1 1cos
2
2. 1 2cos
2
3. 0cos 1
4.
3
4coscos 1
5. 3.0coscos 1
6. 2coscos 1
Domain: ,0 Domain: [-1, 1]
Range: [-1, 1] Range: ,0
The Inverse Tangent Functions:
The inverse Tangent function is the function π‘ππβ1 with domain , and range
2,
2
defined by
tanβ1 π₯ = π¦ β tan π¦ = π₯
The inverse Tangent function is also called the arctangent function, denoted by xarctan .
Evaluate the following expressions
1. 1tan 1 2. 1tan 3 3.
3tantan 1
Evaluate the following expressions
4.
4cossin 1
5.
3
4cossin 1
6.
7
3sincos 1 7. 4tancos 1
8.
5
4cossin 1
Find a simplified expression for each of the following
9.
5cossin 1 x
, for 55 x 10. x3tansin 1
Domain:
2,
2
Domain: ,
Range: , Range:
2,
2
xx
xx
1
1
tantan
tantan
for
for
The cancellation equations are
βπ
2< π₯ <
π
2
x
6.4 Solving Trig Equations Find the solutions of the equation: a) 0 β€ π₯ β€ π; b) 0 β€ π₯ < 2π; c) All Solutions
cos π₯ = β1
2
Solving Trig Equations: To solve a trigonometric equation:
1) Use the rules of algebra to isolate the trigonometric functions on one side of the equal sign.
2) Make a substitution for the inside of the sine or cosine, if it is other then x or ΞΈ.
3) Use our knowledge of the values of the trigonometric functions to solve for the variable:
Use factoring techniques
Use trigonometric identities
4) Use the inverse trig functions to find one solution
5) Use symmetries to find a second solution on one cycle (when a second exists)
6) Find additional solutions if needed by adding full periods
Note: To get all other solutions:
add any integer multiple of 2Ο to solutions related to Sine and Cosine, and
add any integer multiple of Ο to solutions related to Tangent.
7) Undo the substitution
Find all solutions on the interval 0 2
1. 2sin 2 2. 2cos 1
3. sin 1 4. 3tan3
Find all solutions
9. 2cos 2 11. 2sin 1 12. 1tan
Find all solutions on the interval 0 2
13. 2sin 3 1
17. 2cos 2 1
21. cos 14
27. 58.0 sin x
Note: In solving a trig. Equation for aΞΈ, in which the argument is not ΞΈ, like 2sin 3 1 , you must write
down all the solutions first, then solve for ΞΈ .
6.5 Modeling with Trigonometric Equations Solving right triangles In each of the following triangles, solve for the unknown side and angles (in degree).
2.
Modeling with sinusoidal functions Problems that involve quantities that oscillate can often be modeled by a sine or cosine function and once we create a
suitable model for the problem we can use the equation and function values to answer the question.
Find a possible formula for the trigonometric function whose values are in the following tables. 6.
x 0 1 2 3 4 5 6
y 1 -3 -7 -3 1 -3 -7
8. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for
the day is 92 degrees and the low temperature of 78 degrees occurs at 4 AM. Assuming t is the number of hours since
midnight, find an equation for the temperature, D, in terms of t.
B
7
3
A
c
10. A population of elk oscillates 150 above and below an average of 720 during the year, hitting the lowest value in
January (t = 0).
a. Find an equation for the population, P, in terms of the months since January, t.
b. What if the lowest value of the elk population occurred in March instead?
12. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of
84 degrees occurs at 6 PM and the average temperature for the day is 70 degrees. Find the temperature, to the nearest
degree, at 7 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
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