Calculus 1.6

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1997 BC Exam
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1.6 Trig Functions
Black Canyon of the Gunnison National Park,
Colorado
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Answer as quickly as you can!
First, a little review.
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Answer as quickly as you can!
First, a little review.
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Trigonometric functions are used extensively in
calculus.
When you use trig functions in calculus, you must
use radian measure for the angles.
To check or change the angle mode
Press
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.
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If you want to brush up on trig functions, they
are graphed in your book.
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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 doesnt change.
Secant is also an even function, because it is
the reciprocal of cosine.
Even functions are symmetric about the y - axis.
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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.
Cosecant, tangent and cotangent are also odd,
because their formulas contain the sine function.
Odd functions have origin symmetry.
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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
Vertical shift
Positive d moves up.
Horizontal shift
Horizontal stretch or shrink reflection about
y-axis
Positive c moves left.
The horizontal changes happen in the opposite
direction to what you might expect.
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When we apply these rules to sine and cosine, we
use some different terms.
Vertical shift
Horizontal shift
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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.
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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!
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The easiest trig identity is the Pythagorean
Identity
Since the hypotenuse of this triangle has a
length of one, we can just use the Pythagorean
Theorem
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Consider angles u and v in standard position on
the unit circle, determining points A and B and
their coordinates
We could find the length of chord AB by using the
distance formula
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We could rotate angle AOB around to standard
position without changing the length of chord AB
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We could rotate angle AOB around to standard
position without changing the length of chord AB
Using the distance formula
Since the lengths of the chords are the same, we
can set the two expressions equal to each other.
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Starting from this formula we can find a similar
identity
Cosine is an even function, and sine is an odd
function
For convenience, we combine the two formulas like
this
These symbols must be written correctly!
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The co-function identities are simple to find
from the triangle
For example
The co-function identities are not actually
included on the calculus quizzes, but they are
useful.
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Using the properties of odd and even functions
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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 havent already you should
make flashcards and get started memorizing!
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