5
Trigonometric
Identities
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5.5-1
5 Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
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5.5-2
5.5 Double-Angle Identities
Double-Angle Identities ▪ An Application ▪ Product-to-Sum and
Sum-to-Product Identities
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5.5-3
Double-Angle Identities
We can use the cosine sum identity to derive
double-angle identities for cosine.
Cosine sum identity
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5.5-4
Double-Angle Identities
There are two alternate forms of this identity.
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5.5-5
Double-Angle Identities
We can use the sine sum identity to derive a
double-angle identity for sine.
Sine sum identity
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5.5-6
Double-Angle Identities
We can use the tangent sum identity to derive a
double-angle identity for tangent.
Tangent sum identity
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5.5-7
Double-Angle Identities
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1.1-8
5.5-8
Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ
and sin θ < 0, find sin 2θ, cos 2θ, and
Given
tan 2θ.
The identity for sin 2θ requires sin θ.
Any of the three
forms may be used.
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Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Now find tan θ and then use the tangent doubleangle identity.
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1.1-10
5.5-10
Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Alternatively, find tan 2θ by finding the quotient of
sin 2θ and cos 2θ.
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Example 2
FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ
Find the values of the six trigonometric functions of θ if
to find sin θ:
Use the identity
θ is in quadrant II, so sin θ is positive.
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1.1-12
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Example 2
FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ (cont.)
Use a right triangle in quadrant II to find the values of
cos θ and tan θ.
Use the Pythagorean
theorem to find x.
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1.1-13
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Example 3
VERIFYING A DOUBLE-ANGLE IDENTITY
Verify that
is an identity.
Quotient identity
Double-angle
identity
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Example 4
SIMPLIFYING EXPRESSION DOUBLEANGLE IDENTITIES
Simplify each expression.
Multiply by 1.
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Example 5
DERIVING A MULTIPLE-ANGLE
IDENTITY
Write sin 3x in terms of sin x.
Sine sum identity
Double-angle identities
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1.1-16
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Example 6
DETERMINING WATTAGE
CONSUMPTION
If a toaster is plugged into a common household
outlet, the wattage consumed is not constant. Instead
it varies at a high frequency according to the model
where V is the voltage and R is a constant that
measure the resistance of the toaster in ohms.*
Graph the wattage W consumed by a typical toaster
with R = 15 and
in the window
[0, .05] by [–500, 2000]. How many oscillations are
there?
*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition,
Prentice-Hall, 1988.)
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Example 6
DETERMINING WATTAGE
CONSUMPTION
There are six oscillations.
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Product-to-Sum Identities
The identities for cos(A + B) and cos(A – B) can be
added to derive a product-to-sum identity for
cosines.
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5.5-19
Product-to-Sum Identities
Similarly, subtracting cos(A + B) from cos(A – B)
gives a product-to-sum identity for sines.
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5.5-20
Product-to-Sum Identities
Using the identities for sin(A + B) and sine(A – B)
gives the following product-to-sum identities.
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5.5-21
Product-to-Sum Identities
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Example 7
USING A PRODUCT-TO-SUM IDENTITY
Write 4 cos 75° sin 25° as the sum or difference of
two functions.
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Sum-to-Product Identities
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Example 8
Write
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USING A SUM-TO-PRODUCT IDENTITY
as a product of two functions.
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