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5
Trigonometric
Identities
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5
Trigonometric Identities
5.1 Fundamental 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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5.1 Trigonometric Identities
Fundamental Identities ▪ Using the Fundamental Identities
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5.1
Example 1 Finding Trigonometric Function Values Given
One Value and the Quadrant (page 191)
If
value.
(a)
and
is in quadrant IV, find each function
In quadrant IV,
is negative.
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5.1
Example 1 Finding Trigonometric Function Values Given
One Value and the Quadrant (cont.)
If
value.
and
is in quadrant IV, find each function
(b)
(c)
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5.1
Example 2 Expressing One Function in Terms of Another
(page 192)
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5.1
Example 3 Rewriting an Expression in Terms of Sine and
Cosine (page 193)
1  tan2 
Write 1  sec 2  in terms of
and
, and then
simplify the expression so that no quotient appear.
1  tan 
1  tan 

2
1  sec  (sec 2   1)
2
2
sec 2 

 tan2 
1
2
cos


sin2 
cos2 
1
 2
2


csc

sin 
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5.3
Sum and Difference Identities for
Cosine
Difference Identity for Cosine ▪ Sum Identity for Cosine ▪
Cofunction Identities ▪ Applying the Sum and Difference
Identities
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5.3
Example 1 Finding Exact Cosine Function Values
(page 206)
Find the exact value of each expression.
(a)
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2009,
2005 Pearson Education, Inc.
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5.3
Example 1 Finding Exact Cosine Function Values
(cont.)
(b)
(c)
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5.3
Example 2 Using Cofunction Identities to Find θ (page 208)
Find an angle θ that satisfies each of the following.
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5.3
Example 3 Reducing cos (A – B) to a Function of a
Single Variable (page 208)
Write cos(90° + θ) as a trigonometric function of θ
alone.
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t (page 209)
Suppose that
, and both s and t
are in quadrant IV. Find cos(s – t).
The Pythagorean theorem gives
Since s is in quadrant IV, y = –8.
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t (cont.)
Use a Pythagorean identity to find the value of cos t.
Since t is in quadrant IV,
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t (cont.)
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5.3
Example 5 Applying the Cosine Difference Identity
to Voltage (page 221)
Because household current is supplied at different
voltages in different countries, international travelers often
carry electrical adapters to connect items they have
brought from home to a power source. The voltage V in a
typical European 220-volt outlet can be expressed by the
function
(a) European generators rotate at precisely 50 cycles per
second. Determine ω for these electric generators.
Each cycle is
radians at 50 cycles per second.
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5.3
Example 5 Applying the Cosine Difference Identity
to Voltage (cont.)
(b) What is the maximum voltage in the outlet?
The maximum value of
is 1.
The maximum voltage in the outlet is
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5.3
Example 5 Applying the Cosine Difference Identity
to Voltage (cont.)
(c) Determine the least positive value of
that the graph of
the graph of
in radians so
is the same as
Using the sum identity for cosine gives
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5.4
Sum and Difference Identities for
Sine and Tangent
Sum and Difference Identities for Sine ▪ Sum and Difference
Identities for Tangent ▪ Applying the Sum and Difference
Identities
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values (page 217)
Find the exact value of each expression.
(a)
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values (cont.)
(b)
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values (cont.)
(c)
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5.4
Example 2 Writing Functions as Expressions Involving
Functions of θ (page 218)
Write each function as an expression involving
functions of θ.
(a)
(b)
(c)
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5.4
Example 3 Finding Function Values and the Quadrant of
A – B (page 218)
Suppose that A and B are angles in standard position
with
and
Find each of the following.
(c) the quadrant of A – B.
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5.4
Example 3 Finding Function Values and the Quadrant of
A – B (cont.)
The identity for sin(A – B) requires sin A, cos A, sin B,
and cos B. The identity for tan(A – B) requires tan A
and tan B. We must find sin A, tan A, cos B and tan B.
Because A is in quadrant III, sin A is negative and
tan A is positive.
Because B is in quadrant IV, cos B is positive and
tan B is negative.
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5.4
Example 3(a) Finding Function Values and the Quadrant
of A – B (cont.)
To find sin A and cos B, use the identity
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5.4
Example 3(b) Finding Function Values and the Quadrant
of A – B (cont.)
To find tan A and tan B, use the identity
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5.4
Example 3(c) Finding Function Values and the Quadrant
of A – B (cont.)
From parts (a) and (b), sin (A − B) < 0 and
tan (A − B) < 0.
The only quadrant in which the values of both the
sine and the tangent are negative is quadrant IV, so
(A − B) is in quadrant IV.
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5.5 Double-Angle Identities
Double-Angle Identities ▪ An Application ▪ Product-to-Sum and
Sum-to-Product Identities
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ (page 225)
The identity for sin 2θ requires cos θ.
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ (cont.)
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ (cont.)
Alternatively, find tan θ and then use the tangent
double-angle identity.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ (page 226)
Find the values of the six trigonometric functions of θ
if
Use the identity
to find sin θ:
θ is in quadrant III, so sin θ is negative.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ (cont.)
Use the identity
to find cos θ:
θ is in quadrant III, so cos θ is negative.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ (cont.)
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5.5
Example 4 Simplifying Expressions Using Double-Angle
Identities (page 227)
Simplify each expression.
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5.5
Example 5 Deriving a Multiple-Angle Identity (page 228)
Write cos 3x in terms of cos x.
Distributive
property.
Distributive
property.
Simplify.
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5.6 Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities
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5.6
Example 1 Using a Half-Angle Identity to Find an Exact
Value (page 234)
Find the exact value of sin 22.5° using the half-angle
identity for sine.
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5.6
Example 2 Using a Half-Angle Identity to Find an Exact
Value (page 234)
Find the exact value of tan 75° using the identity
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5.6 Example 3
Finding Function Values of s Given
2
Information About s (page 234)
The angle associated with
lies in quadrant II since
is positive while
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are negative.
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5.6 Example 3
Finding Function Values of s Given
2
Information About s (cont.)
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5.6
Example 4 Simplifying Expressions Using the Half-Angle
Identities (page 235)
Simplify each expression.
This matches part of the identity for
.
Substitute 8x for A:
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