Transcript Calculus 1.6

1.6 Trig Functions
Greg Kelly, Hanford High School, Richland, Washington
1.6 Trig Functions
The Mean Streak, Cedar Point Amusement Park, Sandusky, OH
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
on page 41.

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: cos     cos  
Secant is also an even function, because it is the reciprocal
of cosine.
Even functions are symmetric about the y - axis.

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.

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.

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   x  C    D
B

Horizontal shift
B is the period.
B
4
A
3
C
2
D
1
-1
0
-1
 2

y  1.5sin   x  1   2
 4

1
2
x
3
4
5

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 on page 45. To save time, we will use only
five points instead of all the data.

Tuning Fork Data
Time:
Pressure:
.00108
.200
.00198 .00289
.771
-.309
.00379
.480
.00108,.00198,.00289,.00379,.00471  L1
2nd
ENTER
{ .00108,.00198,.00289,.00379,.00471
STO
.2,.771, .309,.48,.581  L2
alpha
.00471
.581
}
2nd
L 1
ENTER
ENTER
SinReg L1, L2 ENTER
2nd
MATH
6
Statistics
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
Done
ENTER

ExpReg L1, L2 ENTER
2nd
MATH
6
Statistics
ShowStat
2nd
MATH
3
9
alpha
SinReg
Regressions
L 1
,
alpha
The calculator
should return:
L 2
ENTER
Done
ENTER
6
Statistics
8
ENTER
ShowStat
The calculator gives
you an equation and
y  a sin b  x  c   d
constants:
a  .608
b  2480
c  2.779
d  .268

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

Plot 1
ENTER
ENTER
WINDOW
GRAPH

WINDOW
GRAPH
You could use the
“trace” function to
investigate the pressure
at any given time.

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


2

2
3
2

2
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and
ranges are defined on page 47.
