Download Trigonometric Identities #1

Identities
zAn identity is a statement that remains
true, regardless of the values of the
variables that appear in it.
{For example: x+x=2x — always true.
zA statement which is true for only some
values of the variables is a conditional
equality.
{For example: 2x+5=13 — only true for x=4.
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Trigonometric Identities
zA trigonometric identity is an identity
involving a combination of trigonometric
functions.
zWe have looked at some identities before:
the definitions of tangent, cosecant,
secant, and cotangent in terms of sine and
cosine.
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Quotient and Reciprocal Identities
zQuotient Identities
sin θ
tan θ =
cos θ
cos θ
cot θ =
sin θ
zReciprocal Identities
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sec θ =
cos θ
1
csc θ =
sin θ
1
cot θ =
tan θ
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Trigonometric Identities
zThese identities, and others we will learn,
are useful for simplifying trigonometric
expressions and equations.
zProving that, in an equation, one side is
always* equal to the other is known as
establishing an identity.
* here, “always” means “in the domains of the functions involved”
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Establishing Identities
zEstablishing an identity is just like solving
a fun mathematical jigsaw puzzle!
{One side of the equation consists of the
“pieces” you have to work with.
{The known identities are what you use to
manipulate the pieces to fit together.
{The other side of the equation is the picture on
the box… that is your goal!
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Example #1
zWe want to show that
sin x cot x = cos x
zWe should start on the left side, since it is
more complicated, and make it look like
the right.
zWe can algebraically manipulate either
side of the equation, but can’t move
anything across the equal sign!
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Example #1
zWe know the definition of cot x in terms of
sin x and cos x:
cos x
cot x =
sin x
zConverting to sines and cosines is a good
idea, because, chances are, things will
cancel very nicely.
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Example #1
zTherefore,
sin x cot x
cos x
sin x ·
sin x
cos x
=
cos x
=
cos x
=
cos x
¤
zThe identity has been established.
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Examples #2-3
csc x tan x
sin x
1
·
sin x cos x
1
cos x
sec x
=
sec x
=
sec x
=
sec x
=
sec x ¤
sin2 θ sec θ csc θ
1
1
·
sin θ · sin θ ·
cos θ sin θ
sin θ
cos θ
tan θ
The notation sin2 θ means (sin θ)2 .
= tan θ
= tan θ
= tan θ
= tan θ
¤
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Pythagorean Identities
zOn the unit circle, the sine and cosine of
the angle describe two legs of a right
triangle, while the radius (of 1) is the
hypotenuse.
zThus, it follows from the Pythagorean
theorem that
2
2
sin θ + cos θ = 1
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Pythagorean Identities
zUsing this identity, we can create the other
two Pythagorean identities. (Do you see
how?)
2
2
sin θ + cos θ
2
tan θ + 1
2
1 + cot θ
=
=
=
1
2
sec θ
2
csc θ
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Example #4
zWe want to show that
2
2
2
cos β − sin β = 1 − 2 sin β
zLet’s start on the left side. We can rewrite
the first Pythagorean identity in terms of
one of the given terms to eliminate it.
sin2 β + cos2 β = 1 ⇒ cos2 β = 1 − sin2 β
zAfter we do that, we can combine like
terms and simplify.
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Example #4
zTherefore,
cos2 β − sin2 β
¢
¡
2
1 − sin β − sin2 β
2
1 − 2 sin β
=
=
=
1 − 2 sin2 β
1 − 2 sin2 β
2
1 − 2 sin β
¤
zWe could have eliminated the other term
on the left or either term on the right. All
methods would establish the identity.
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Example #5
zWe want to show that
2
(1 + cos x)(1 − cos x) = sin x
zTo establish this identity, we can FOIL the
left side and then cancel terms.
(1 + cos x)(1 − cos x)
1 − cos x + cos x − cos2 x
1 − cos2 x
2
sin x
=
=
=
=
sin2 x
sin2 x
sin2 x
2
sin x ¤
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Example #6
zWe want to show that
cot x + tan x = csc x sec x
zFirst, let’s rewrite everything in terms of
sines and cosines.
cos x
sin x
1
1
+
=
·
sin x
cos x
sin x cos x
zTo add, we need a common denominator,
which in this case is sin x cos x .
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Example #6
zTherefore,
sin x
cos x
+
sin x ¶ cos x
µ
³ cos x ´ cos x
sin x sin x
+
sin x cos x
cos x sin x
cos2 x
sin2 x
+
sin x cos x sin x cos x
cos2 x + sin2 x
sin x cos x
1
sin x cos x
=
=
=
=
=
1
1
·
sin x cos x
1
sin x cos x
1
sin x cos x
1
sin x cos x
1
sin x cos x
¤
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Example #7
zWe want to show that
sin θ
1 − cos θ
=
sin θ
1 + cos θ
zTo establish this identity, we take the more
complicated right side (since it has
addition in the denominator) and multiply
by the conjugate of the denominator.
zRemember, you cannot cross multiply!
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Example #7
zThe conjugate of a binomial takes the
original two factors and reverses the sign
in between. The conjugate of a+b is a–b.
zIn this example, the conjugate of 1 + cos θ
is 1 − cos θ .
zMultiplying the right side by a fancy form of
1 involving this conjugate will create
something that FOILS and cancels nicely.
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Example #7
zTherefore,
1 − cos θ
sin θ
1 − cos θ
sin θ
1 − cos θ
sin θ
1 − cos θ
sin θ
1 − cos θ
sin θ
=
=
=
=
=
sin θ
1 + cos θ
sin θ
1 − cos θ
·
1 + cos θ 1 − cos θ
sin θ(1 − cos θ)
1 − cos2 θ
sin θ(1 − cos θ)
sin2 θ
1 − cos θ
sin θ
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