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All of the identities we learned are found on the back page of your book.
You'll need to have these memorized or be able to derive them for this course.
QUOTIENT IDENTITIESsin
tancos
x
x
x
cos
cotsin
x
x
x
2 2tan 1 secx x
2 21 cot cscx x PYTHAGOREAN IDENTITIES
2 2sin cos 1x x
RECIPROCAL IDENTITIES
1csc
sinx
x
1
seccos
x
x
1
cottan
x
x
1sin
cscx
x
1
cossec
x
x
1
tancot
x
x
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One way to use identities is to simplify expressions
involving trigonometric functions. Often a good strategy for
doing this is to write all trig functions in terms of sines and
cosines and then simplify. Lets see an example of this:
sintan
cos
x
x
x
1sec
cosx
x
1csc
sinx
x
tan cscSimplify:sec
x x
x
sin 1
cos sin1
cos
x
x x
x
substitute using
each identity
simplify
1
cos1
cos
x
x
1
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Another way to use identities is to write one function in
terms of another function. Lets see an example of this:
2
Write the following expressionin terms of only one trig function:
cos sin 1x x This expression involves both
sine and cosine. The
Fundamental Identity makes a
connection between sine and
cosine so we can use that and
solve for cosine squared and
substitute.
2 2sin cos 1x x 2 2cos 1 sinx x
2= 1 sin
sin 1x x
2= sin sin 2x x
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A third way to use identities is to find function values. Lets
see an example of this:
2
Write the following expressionin terms of only one trig function:
cos sin 1x x This expression involves both
sine and cosine. The
Fundamental Identity makes a
connection between sine and
cosine so we can use that and
solve for cosine squared and
substitute.
2 2sin cos 1x x 2 2cos 1 sinx x
2= 1 sin sin 1x x
2= sin sin 2x x
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1Given sin with in quadrant II,3
find the other five trig functions using identities.
We'd get csc by takingreciprocal of sin
csc 3
Now use the fundamental trig identity
1cossin22
Sub in the value of sine that you know
1cos3
1 22
Solve this for cos
9
8cos
2 8 2 2
cos39
When we square root, we need but determine that wed
need the negative since we have an angle in Quad II wherecosine values are negative.
square root
both sides
A third way to use identities is to find function values. Lets
see an example of this: 1csc
sin
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2 2cos
3
3
1sin
csc 3
We need to get tangent using
fundamental identities.
cos
sintan
Simplify by inverting and multiplying1
3tan 2 2
3
Finally you can find
cotangent by taking the
reciprocal of this answer.
3sec
2 2
1 3
3 2 2
1
2 2
cot 2 2
You can easily find sec by taking reciprocal of cos.
This can be rationalized
2
23 2
4
24
This can be rationalized
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Now lets look at the unit circle to compare trig functions
of positive vs. negative angles.
?3
cosisWhat
?3
cosisWhat
Remember a negative
angle means to go
clockwise
2
1
2
1
2
3,
2
1
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cos cosx x Recall from College Algebra that if we put
a negative in the function and get the
original back it is an even function.
?3
sinisWhat
?3
sinisWhat
2
3
2
3
2
3,
2
1
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sin sinx x Recall from College Algebra that if we
put a negative in the function and get
the negative of the function back it is an
odd function.
?3
tanisWhat
?3
tanisWhat
2
3,
2
1
3
3
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If a function is even, its reciprocal function will be
also. If a function is odd its reciprocal will be also.
EVEN-ODD PROPERTIES
sin(- x ) = - sinx (odd) csc(- x ) = - cscx(odd)
cos(- x) = cos x (even) sec(- x ) = sec x (even)
tan(- x) = - tan x (odd) cot(- x ) = - cot x (odd)
angle?positiveaoftermsinwhat60sin
60sin
angle?positiveaoftermsinwhat3
2sec
3
2
sec
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RECIPROCAL IDENTITIES1
cscsin
x
x
1
seccos
x
x
1cottan
x
x
QUOTIENT IDENTITIESsin
tancos
x
x
x
cos
cotsin
x
x
x
2 2tan 1 secx x 2 2
1 cot cscx x
PYTHAGOREAN IDENTITIES
2 2sin cos 1x x
EVEN-ODD IDENTITIES
sin sin cos cos tan tan
csc csc sec sec cot cot
x x x x x x
x x x x x x
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