1997 BC Exam

1.6 Trig Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

Black Canyon of the GunnisonNational Park, Colorado

x

y

0

2

3

2

2

First, a little review.

Answer as quickly as you can!

3tan

4

1

cot undefined

3csc

4

2

tan4

1

5sec

4

2

3cos

4

1

2

x

y

0

2

3

2

2

First, a little review.

Answer as quickly as you can!

3sin

4

1

2

3sec

4

2

7cos

4

1

2

3sin

2

1

5csc

4

2

sin 0 0

To check or change the angle mode:

When you use trig functions in calculus, you must use radian measure for the angles.

Press: 5

Settings

2

Document Settings

Trigonometric functions are used extensively in calculus.

Make sure you set the angle mode to Radian, then scroll down and click Make Default.

You could also click Restore, which returns the calculator to the factory settings, which include radian mode, and then click Make Default.

The best plan is to leave the calculator mode to radians and use when you need to use degrees.o

If you want to brush up on trig functions, they are graphed in your book.

To find trig functions on the TI-nspire, press , select the desired function, and press . enter

trig

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

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 in the book.

siny x

You will be using trig identities throughout the year to solve calculus problems.

Today we will look at some of those identities and where they come from.

When you need to use a trig identity you will not have time to generate the identity from scratch. They need to be memorized!

The easiest trig identity is the Pythagorean Identity:

x

y

A cos ,sin

O

Since the hypotenuse of this triangle has a length of one, we can just use the Pythagorean Theorem:

2 2sin cos 1

1,0

Consider angles u and v in standard position on the unit circle, determining points A and B and their coordinates:

cos ,sinu u 2 2cos cos sin sinAB v u v u

x

y

u

v

A

B cos ,sinv v We could find the length of

chord AB by using the distance formula:

Let the difference between the angles be: u v

O

cos ,sinu u

x

y

B cos ,sinv v

O

A

We could rotate angle AOB around to standard position without changing the length of chord AB:

2 2cos cos sin sinAB v u v u

We could rotate angle AOB around to standard position without changing the length of chord AB:

1,0x

y

A

B

cos ,sin

We rewrite the coordinates of A and B in terms of :

O

Using the distance formula:

2 2cos 1 sin 0AB

Since the lengths of the chords are the same, we can set the two expressions equal to each other.

2 2 2 2cos cos sin sin cos 1 sin 0v u v u

2 2 2 2cos cos sin sin cos 1 sinv u v u

2 2 2 2cos 2cos cos cos sin 2sin sin sinv u v u v u v u 2 2cos 2cos 1 sin

1 2cos cos 1 2sin sin 1 2cos 1u v u v

2 2cos cos 2sin sin 2 2cosu v u v

1 cos cos sin sin 1 cosu v u v

cos cos sin sin cosu v u v

cos cos cos sin sinu v u v

cos cos cos sin sinu v u v u v u v

cos cos cos sin sinu v u v u v

Starting from this formula we can find a similar identity:

cos cos vu v u

cos cos cos sin sinu v u v vu

Cosine is an even function, and sine is an odd function:

cos cos cos sin sinu v u v u v

For convenience, we combine the two formulas like this:

cos cos cos sin sinu v u v u v

These symbols must be written correctly!

2

x

yz

The co-function identities are simple to find from the triangle:

oppositesin

hypotenuse

y

z adjacent

cos2 hypotenuse

y

z

sin cos2

For example:

The co-function identities are not actually included on the calculus quizzes, but they are useful.

sin cos2

sin cos2

u v u v

sin cos2

u v u v cos cos cos sin sinu v u v u v

sin cos cos sin sin2 2

u v u v u v

sin sin cos cos sinu v u v u v

sin sin cos cos sinu v u v u v Using the properties of odd and even functions:

There are sixteen trig identities on the calculus formula sheets.

Starting with the formulas in this lecture, you should be able to derive the others for practice, or for fun!

These formulas are sometimes difficult to remember, so if you haven’t already you should make flashcards and get started memorizing!