1.6 Trig Functions Black Canyon of the Gunnison National Park, Colorado Trigonometric functions are used extensively in calculus. When you use trig functions.
Download ReportTranscript 1.6 Trig Functions Black Canyon of the Gunnison National Park, Colorado Trigonometric functions are used extensively in calculus. When you use trig functions.
Slide 1
1.6 Trig Functions
Black Canyon of the Gunnison
National Park, Colorado
Slide 2
Trigonometric functions are used extensively in calculus.
When you use trig functions in calculus, you must use radian
measure for the angles. The best plan is to set the calculator
o when you need to use
mode to radians and use 2nd
degrees.
If you want to brush up on trig functions, they are graphed in
your book.
Slide 3
Even and Odd Trig Functions:
“Even” functions behave like polynomials with even
exponents, in that when you change the sign of x, the y
value doesn’t change.
Cosine is an even function because: co s co s
Secant is also an even function, because it is the reciprocal
of cosine.
Even functions are symmetric about the y - axis.
Slide 4
Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd
exponents, in that when you change the sign of x, the
sign of the y value also changes.
Sine is an odd function because:
sin
sin
Cosecant, tangent and cotangent are also odd, because
their formulas contain the sine function.
Odd functions have origin symmetry.
Slide 5
The rules for shifting, stretching, shrinking, and reflecting the
graph of a function apply to trigonometric functions.
Vertical stretch or shrink;
reflection about x-axis
a 1 is a stretch.
Vertical shift
Positive d moves up.
y a f b x c d
Horizontal shift
Horizontal stretch or shrink;
Positive c moves left.
reflection about y-axis
b 1 is a shrink. The horizontal changes happen
in the opposite direction to what
you might expect.
Slide 6
When we apply these rules to sine and cosine, we use some
different terms.
A is the amplitude.
Vertical shift
2
f x A sin
B
Horizontal shift
B is the period.
B
4
A
3
x C D
C
2
D
1
-1
0
-1
2
y 1.5 sin
4
1
2
x
3
4
x 1 2
5
Slide 7
Trig functions are not one-to-one.
However, the domain can be restricted for trig functions
to make them one-to-one.
2
y sin x
3
2
3
2
2
2
2
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and
ranges are defined in the book.
*
Slide 8
The sine equation is built into the TI-89 as a
sinusoidal regression equation.
For practice, we will find the sinusoidal equation for the
tuning fork data in the book. To save time, we will use only
five points instead of all the data.
Slide 9
Tuning Fork Data
Time:
Pressure:
.00108
.200
.00198 .00289
.771
-.309
.0 0 1 0 8, .0 0 1 9 8, .0 0 2 8 9, .0 0 3 7 9, .0 0 4 7 1
2nd
.00379
.480
L1
ENTER
{ .00108,.00198,.00289,.00379,.00471
STO
.2, .7 7 1, .3 0 9, .4 8, .5 8 1
L2
.00471
.581
alpha
}
2nd
L 1
ENTER
ENTER
S inR eg L1, L 2 E N T E R
2nd
MATH
6
Statistics
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
Slide 10
E xpR eg L1, L 2 E N T E R
2nd
MATH
6
Statistics
S h o w S tat
2nd
MATH
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
ENTER
6
Statistics
8
ENTER
ShowStat
a .608
The calculator gives
you an equation and
y a sin b x c d
constants:
b 2480
c 2.779
d .268
Slide 11
We can use the calculator to plot the new curve along with
the original points:
Y=
2nd
Plot 1
y1=regeq(x)
VAR-LINK
x
)
regeq
ENTER
ENTER
WINDOW
Slide 12
Plot 1
ENTER
ENTER
WINDOW
GRAPH
Slide 13
WINDOW
GRAPH
You could use the
“trace” function to
investigate the pressure
at any given time.
1.6 Trig Functions
Black Canyon of the Gunnison
National Park, Colorado
Slide 2
Trigonometric functions are used extensively in calculus.
When you use trig functions in calculus, you must use radian
measure for the angles. The best plan is to set the calculator
o when you need to use
mode to radians and use 2nd
degrees.
If you want to brush up on trig functions, they are graphed in
your book.
Slide 3
Even and Odd Trig Functions:
“Even” functions behave like polynomials with even
exponents, in that when you change the sign of x, the y
value doesn’t change.
Cosine is an even function because: co s co s
Secant is also an even function, because it is the reciprocal
of cosine.
Even functions are symmetric about the y - axis.
Slide 4
Even and Odd Trig Functions:
“Odd” functions behave like polynomials with odd
exponents, in that when you change the sign of x, the
sign of the y value also changes.
Sine is an odd function because:
sin
sin
Cosecant, tangent and cotangent are also odd, because
their formulas contain the sine function.
Odd functions have origin symmetry.
Slide 5
The rules for shifting, stretching, shrinking, and reflecting the
graph of a function apply to trigonometric functions.
Vertical stretch or shrink;
reflection about x-axis
a 1 is a stretch.
Vertical shift
Positive d moves up.
y a f b x c d
Horizontal shift
Horizontal stretch or shrink;
Positive c moves left.
reflection about y-axis
b 1 is a shrink. The horizontal changes happen
in the opposite direction to what
you might expect.
Slide 6
When we apply these rules to sine and cosine, we use some
different terms.
A is the amplitude.
Vertical shift
2
f x A sin
B
Horizontal shift
B is the period.
B
4
A
3
x C D
C
2
D
1
-1
0
-1
2
y 1.5 sin
4
1
2
x
3
4
x 1 2
5
Slide 7
Trig functions are not one-to-one.
However, the domain can be restricted for trig functions
to make them one-to-one.
2
y sin x
3
2
3
2
2
2
2
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and
ranges are defined in the book.
*
Slide 8
The sine equation is built into the TI-89 as a
sinusoidal regression equation.
For practice, we will find the sinusoidal equation for the
tuning fork data in the book. To save time, we will use only
five points instead of all the data.
Slide 9
Tuning Fork Data
Time:
Pressure:
.00108
.200
.00198 .00289
.771
-.309
.0 0 1 0 8, .0 0 1 9 8, .0 0 2 8 9, .0 0 3 7 9, .0 0 4 7 1
2nd
.00379
.480
L1
ENTER
{ .00108,.00198,.00289,.00379,.00471
STO
.2, .7 7 1, .3 0 9, .4 8, .5 8 1
L2
.00471
.581
alpha
}
2nd
L 1
ENTER
ENTER
S inR eg L1, L 2 E N T E R
2nd
MATH
6
Statistics
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
Slide 10
E xpR eg L1, L 2 E N T E R
2nd
MATH
6
Statistics
S h o w S tat
2nd
MATH
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
ENTER
6
Statistics
8
ENTER
ShowStat
a .608
The calculator gives
you an equation and
y a sin b x c d
constants:
b 2480
c 2.779
d .268
Slide 11
We can use the calculator to plot the new curve along with
the original points:
Y=
2nd
Plot 1
y1=regeq(x)
VAR-LINK
x
)
regeq
ENTER
ENTER
WINDOW
Slide 12
Plot 1
ENTER
ENTER
WINDOW
GRAPH
Slide 13
WINDOW
GRAPH
You could use the
“trace” function to
investigate the pressure
at any given time.