6.1 Sinusoidal Graphs - Amazon S3โฌยฆย ยท In Function Notation, periodic means that, for all x in...
Transcript of 6.1 Sinusoidal Graphs - Amazon S3โฌยฆย ยท In Function Notation, periodic means that, for all x in...
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,
๐(๐ฅ + ๐) = ๐(๐ฅ) 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 functionโs maximum and minimum
values.
The amplitude is the vertical distance between the functionโs 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)
๐ก 0 ๐
4
๐
2
3๐
4 ๐
5๐
4
3๐
2
7๐
4 2๐
sin ๐ก
cos ๐ก
Period: _________
Midline: ________
Amplitude: ______
Period: _________
Midline: ________
Amplitude: ______
Amplitude
๐ฆ = ๐ sin ๐ก,
then |๐| is the amplitude of the graph.
Ex. ๐ฆ = sin ๐ก
๐ฆ = 2 sin ๐ก
๐ฆ = โ sin ๐ก
Midline
The graph of ๐ฆ = ๐(๐ก) + ๐ is the graph of ๐ฆ = ๐(๐ก) shifted vertically by d units.
Note: If ๐ฆ = sin ๐ก + ๐ and ๐ฆ = cos ๐ก + ๐, we have midlines ๐ฆ = ๐.
Ex. ๐ฆ = cos ๐ก + 2
Note:
Midline of both graphs is ๐ฆ = 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 = |๐ด|
period, B
P2
midline ๐ฆ = ๐
h is the horizontal shift of the function
An appropriate interval on which to graph one complete period is[โ, โ + (2๐๐ตโ )]
Note: horizontal shift means: the curves shifted horizontally by an amount |โ|. They are shifted to the right if h >
0 or to the left if h < 0.
Period
Ex. ๐ฆ = sin(2๐ก) Ex. ๐ฆ = sin(๐ก2โ )
Horizontal Shift
Ex. ๐ฆ = 2sin(๐ก + ๐2โ ) Ex. ๐ฆ = โ3sin(2๐ก โ ๐
2โ )
For the equation below, determine the amplitude, midline, period, and horizontal shift (indicated the direction).
๐ฆ = โ5sin(๐๐ก + 2) โ 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
๐ฅ = and ๐ฆ =
Find the slope of the line ๐๐ ,
Graph of the Tangent Function (๐ = ๐๐๐๐ฝ)
Period =
Range:
Domain:
Period =
Range:
Domain:
P(x, y)
1
ฮธ y
x 0
Note: The functions ๐ฆ = ๐ด tan(๐ต๐ฅ) and ๐ฆ = ๐ด cot(๐ต๐ฅ), where ๐ต > 0, have period ๐
๐ต.
Graph of ๐ฆ = csc ๐ฅ Graph of ๐ฆ = sec ๐ฅ
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 ๐ฆ = ๐ด csc(๐ต๐ฅ) and ๐ฆ = ๐ด sec(๐ต๐ฅ), where ๐ต > 0, have period 2๐
๐ต.
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
๐โ1(๐ฅ) = ๐ฆ โ ๐(๐ฆ) = ๐ฅ Therefore,
f-1
is the rule that reverse the action of f. (From output back to input)
Note: ๐โ1 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 ๐ ๐๐โ1 with domain [-1, 1] and range
2,
2
defined by
sinโ1 ๐ฅ = ๐ฆ โ sin ๐ฆ = ๐ฅ
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 ๐๐๐ โ1 with domain [-1, 1] and range ,0 defined by
cosโ1 ๐ฅ = ๐ฆ โ cos ๐ฆ = ๐ฅ
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 ๐ก๐๐โ1 with domain , and range
2,
2
defined by
tanโ1 ๐ฅ = ๐ฆ โ tan ๐ฆ = ๐ฅ
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
โ๐
2< ๐ฅ <
๐
2
x
6.4 Solving Trig Equations Find the solutions of the equation: a) 0 โค ๐ฅ โค ๐; b) 0 โค ๐ฅ < 2๐; c) All Solutions
cos ๐ฅ = โ1
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 ฮธ.
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 2ฯ to solutions related to Sine and Cosine, and
add any integer multiple of ฯ 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 aฮธ, in which the argument is not ฮธ, like 2sin 3 1 , you must write
down all the solutions first, then solve for ฮธ .
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.