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 mode to radians and use when you need to use degrees.

2nd o

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.

y a f b x c d

Vertical stretch or shrink;reflection about x-axis

Horizontal stretch or shrink;reflection about y-axis

Horizontal shift

Vertical shift

Positive c moves left.

Positive d moves up.

The horizontal changes happen in the opposite direction to what you might expect.

is a stretch.1a

is a shrink.1b

When we apply these rules to sine and cosine, we use some different terms.

2sinf x A x C D

B

Horizontal shift

Vertical shift

is the amplitude.A

is the period.B

A

B

C

D 21.5sin 1 2

4y x

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.

Time: .00108 .00198 .00289 .00379 .00471 Pressure: .200 .771 -.309 .480 .581

.00108,.00198,.00289,.00379,.00471 L1 ENTER

2nd { .00108,.00198,.00289,.00379,.00471 2nd }

STO alpha L 1 ENTER

.2,.771, .309,.48,.581 L2 ENTER

SinReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

9

SinReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

Tuning Fork Data

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants: siny a b x c d

.608

2480

2.779

.268

a

b

c

d

2nd MATH 6 3

Statistics Regressions

9

SinReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

ExpReg L1, L2 ENTER

We can use the calculator to plot the new curve along with the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 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.

2 3

2

2

2

3

2

2

Trig functions are not one-to-one.

However, the domain can be restricted for trig functions to make them one-to-one.

These restricted trig functions have inverses.

Inverse trig functions and their restricted domains and ranges are defined on page 47.

siny x