An identity is an equation that is true for all defined values of a variable.
We are going to use the identities that we have already established and establish others to "prove" or verify other identities. Let's summarize the basic identities we have.
RECIPROCAL IDENTITIES
1csc
sinx
x
1sec
cosx
x
1cot
tanx
x
QUOTIENT IDENTITIESsin
tancos
xx
x
coscot
sin
xx
x
1sin
cscx
x
1cos
secx
x
1tan
cotx
x
2 2sin cos 1x x Let’s look at the Fundamental Identity derived in Section 1.6
Now to find the two more identities from this famous and oft used one.
2 2sin cos 1x x Divide all terms by cos2x
cos2x cos2x cos2xWhat trig function is this squared? 1 What trig function
is this squared?
2 2tan 1 secx x 2 2sin cos 1x x Divide all terms by sin2x
sin2x sin2x sin2x
What trig function is this squared?
1 What trig function is this squared?
2 21 cot cscx x
These three are sometimes called the Pythagorean Identities since the derivation of the fundamental theorem used the Pythagorean Theorem
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
xx
x
coscot
sin
xx
x
2 2tan 1 secx x
2 21 cot cscx x PYTHAGOREAN IDENTITIES
2 2sin cos 1x x
RECIPROCAL IDENTITIES
1csc
sinx
x
1sec
cosx
x
1cot
tanx
x
1sin
cscx
x
1cos
secx
x
1tan
cotx
x
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. Let’s see an example of this:
sintan
cos
xx
x
1sec
cosx
x
1csc
sinx
x
tan cscSimplify:
sec
x x
x
sin 1cos sin
1cos
xx x
x
substitute using each identity
simplify1
cos1
cos
x
x
1
Another way to use identities is to write one function in terms of another function. Let’s see an example of this:
2
Write the following expression
in 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
1Given sin with in quadrant II,
3find the other five trig functions using identities.
We'd get csc by taking reciprocal of sin
csc 3Now use the fundamental trig identity1cossin 22
Sub in the value of sine that you know
1cos3
1 22
Solve this for cos
9
8cos2
8 2 2cos
39
When we square root, we need but determine that we’d need the negative since we have an angle in Quad II where cosine values are negative.
square root both sides
A third way to use identities is to find function values. Let’s see an example of this: 1
cscsin
2 2cos
3
3
1sin
csc 3
We need to get tangent using fundamental identities.
cos
sintan
Simplify by inverting and multiplying13tan
2 23
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
2 3 2
4
2
4
This can be rationalized
Now let’s look at the unit circle to compare trig functions of positive vs. negative angles.
?3
cos isWhat
?3
cos isWhat
Remember a negative angle means to go clockwise
2
1
2
1
2
3,
2
1
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
sin isWhat
?3
sin isWhat
2
3
2
3
2
3,
2
1
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
If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also.
EVEN-ODD PROPERTIESsin(- x ) = - sin x (odd) csc(- x ) = - csc x (odd)
cos(- x) = cos x (even) sec(- x ) = sec x (even)
tan(- x) = - tan x (odd) cot(- x ) = - cot x (odd)
angle? positivea of termsin what 60sin
60sin
angle? positivea of termsin what 3
2sec
3
2sec
RECIPROCAL IDENTITIES1
cscsin
xx
1
seccos
xx
1cot
tanx
x
QUOTIENT IDENTITIESsin
tancos
xx
x
coscot
sin
xx
x
2 2tan 1 secx x 2 21 cot cscx x
PYTHAGOREAN IDENTITIES2 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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