Copyright © 2009 Pearson Addison-Wesley 5.1-2
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
5 Trigonometric Identities
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Fundamental Identities5.1Fundamental Identities Using the Fundamental Identities
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Fundamental Identities
Reciprocal Identities
Quotient Identities
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Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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Note
In trigonometric identities, θ can be
an angle in degrees, an angle in
radians, a real number, or a variable.
Copyright © 2009 Pearson Addison-Wesley 1.1-75.1-7
If and θ is in quadrant II, find each function
value.
Example 1 FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT
(a) sec θ
In quadrant II, sec θ is negative, so
Pythagorean identity
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Example 1 FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) sin θ
from part (a)
Quotient identity
Reciprocal identity
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Example 1 FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) cot(– θ) Reciprocal identity
Negative-angle identity
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Caution
To avoid a common error, when
taking the square root, be sure to
choose the sign based on the
quadrant of θ and the function being
evaluated.
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Express cos x in terms of tan x.
Example 2 EXPRESSING ONE FUNCITON IN
TERMS OF ANOTHER
Since sec x is related to both cos x and tan x by
identities, start with
Take reciprocals.
Reciprocal identity
Take the square root of each side.
The sign depends on the quadrant of x.
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Write tan θ + cot θ in terms of sin θ and cos θ, and
then simplify the expression.
Example 3 REWRITING AN EXPRESSION IN
TERMS OF SINE AND COSINE
Quotient identities
Write each fraction with the LCD.
Pythagorean identity
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Caution
When working with trigonometric
expressions and identities, be sure
to write the argument of the function.
For example, we would not write
An argument such as θ
is necessary.
Copyright © 2009 Pearson Addison-Wesley 5.2-2
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
5 Trigonometric Identities
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Verifying Trigonometric
Identities5.2
Verifying Identities by Working With One Side ▪ Verifying
Identities by Working With Both Sides
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Hints for Verifying Identities
Learn the fundamental identities.
Whenever you see either side of a fundamental identity, the other side should come to mind. Also, be aware of equivalent forms of the fundamental identities.
Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.
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Hints for Verifying Identities
It is sometimes helpful to express all trigonometric functions in the equation in terms of sine and cosine and then simplify the result.
Usually, any factoring or indicated algebraic operations should be performed.
For example, the expression
can be factored as
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Hints for Verifying Identities
The sum or difference of two trigonometric
expressions such as can be
added or subtracted in the same way as
any other rational expression.
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Hints for Verifying Identities
As you select substitutions, keep in mind the side you are not changing, because it represents your goal.
For example, to verify the identity
find an identity that relates tan x to cos x.
Since and
the secant function is the best link between
the two sides.
Copyright © 2009 Pearson Addison-Wesley 1.1-85.2-8
Hints for Verifying Identities
If an expression contains 1 + sin x, multiplying both the numerator and denominator by 1 – sin x would give 1 – sin2 x, which could be replaced with cos2x.
Similar results for 1 – sin x, 1 + cos x, and 1 – cos x may be useful.
Copyright © 2009 Pearson Addison-Wesley 1.1-95.2-9
Caution
Verifying identities is not the same a
solving equations.
Techniques used in solving equations,
such as adding the same terms to both
sides, should not be used when
working with identities since you are
starting with a statement that may not
be true.
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Verifying Identities by Working with One Side
To avoid the temptation to use algebraic properties
of equations to verify identities, one strategy is to
work with only one side and rewrite it to match
the other side.
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Example 1 VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that is an identity.
Work with the right side since it is more complicated.
Right side of given
equation
Distributive
property
Left side of given
equation
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Example 2 VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that is an identity.
Distributive
property
Left side
Right side
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Example 3 VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that is an identity.
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Example 4 VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that is an identity.
Multiply by 1
in the form
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Verifying Identities by Working with Both Sides
If both sides of an identity appear to be equally
complex, the identity can be verified by working
independently on each side until they are changed
into a common third result.
Each step, on each side, must be reversible.
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Example 5 VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES)
Verify that is an
identity.
Working with the left side: Multiply by 1
in the form
Distributive
property
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Example 5 VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES) (continued)
Working with the right side:
Factor the numerator.
Factor the
denominator.
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Example 5 VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES) (continued)
So, the identity is verified.
Right side of given
equation
Left side of given
equation
Common third
expression
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Example 6 APPLYING A PYTHAGOREAN IDENTITY
TO RADIOS
Tuners in radios select a radio station by adjusting
the frequency. A tuner may contain an inductor L and
a capacitor. The energy stored in the inductor at time
t is given by
and the energy in the capacitor is given by
where f is the frequency of the radio station and k is a
constant.
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Example 6 APPLYING A PYTHAGOREAN IDENTITY
TO RADIOS (continued)
The total energy in the circuit is given by
Show that E is a constant function.*
*(Source: Weidner, R. and R. Sells, Elementary Classical Physics, Vol. 2,
Allyn & Bacon, 1973.)
Copyright © 2009 Pearson Addison-Wesley 1.1-215.2-21
Example 6 APPLYING A PYTHAGOREAN IDENTITY
TO RADIOS (continued)
Factor.
Copyright © 2009 Pearson Addison-Wesley 5.2-2
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
5 Trigonometric Identities
Copyright © 2009 Pearson Addison-Wesley 1.1-35.2-3
Sum and Difference
Identitites for Cosine5.3
Difference Identity for Cosine ▪ Sum Identity for Cosine ▪
Cofunction Identities ▪ Applying the Sum and Difference Identities
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Difference Identity for Cosine
Point Q is on the unit
circle, so the coordinates
of Q are (cos B, sin B).
The coordinates of S are
(cos A, sin A).
The coordinates of R are (cos(A – B), sin (A – B)).
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Difference Identity for Cosine
Since the central angles
SOQ and POR are
equal, PR = SQ.
Using the distance formula,
we have
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Difference Identity for Cosine
Square both sides and clear parentheses:
Rearrange the terms:
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Difference Identity for Cosine
Subtract 2, then divide by –2:
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Sum Identity for Cosine
To find a similar expression for cos(A + B) rewrite
A + B as A – (–B) and use the identity for
cos(A – B).
Cosine difference identity
Negative angle identities
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Example 1(a) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of cos 15 .
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Example 1(b) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of
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Example 1(c) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of cos 87 cos 93 – sin 87 sin 93 .
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Cofunction Identities
Similar identities can be obtained for a real number domain by replacing 90with
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Example 2 USING COFUNCTION IDENTITIES TO
FIND θ
Find an angle that satisfies each of the following:
(a) cot θ = tan 25
(b) sin θ = cos (–30 )
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Example 2 USING COFUNCTION IDENTITIES TO
FIND θ
Find an angle that satisfies each of the following:
(c)
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Note
Because trigonometric (circular)
functions are periodic, the solutions
in Example 2 are not unique. Only
one of infinitely many possiblities
are given.
Copyright © 2009 Pearson Addison-Wesley 5.2-17
Applying the Sum and Difference
Identities
If one of the angles A or B in the identities for
cos(A + B) and cos(A – B) is a quadrantal angle,
then the identity allows us to write the expression
in terms of a single function of A or B.
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Example 3 REDUCING cos (A – B) TO A FUNCTION
OF A SINGLE VARIABLE
Write cos(90 + θ) as a trigonometric function of θ
alone.
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Example 4 FINDING cos (s + t) GIVEN
INFORMATION ABOUT s AND t
Suppose that and both s and t
are in quadrant II. Find cos(s + t).
Sketch an angle s in quadrant II
such that Since
let y = 3 and r = 5.
The Pythagorean theorem gives
Since s is in quadrant II, x = –4 and
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Example 4 FINDING cos (s + t) GIVEN
INFORMATION ABOUT s AND t (cont.)
Sketch an angle t in quadrant II
such that Since
let x = –12 and
r = 5.
The Pythagorean theorem gives
Since t is in quadrant II, y = 5 and
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Example 4 FINDING cos (s + t) GIVEN
INFORMATION ABOUT s AND t (cont.)
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Note
The values of cos s and sin t could
also be found by using the
Pythagorean identities.
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Example 5 APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE
Common household current is called alternating
current because the current alternates direction
within the wires. The voltage V in a typical 115-volt
outlet can be expressed by the function
where ω is the angular speed (in radians per second)
of the rotating generator at the electrical plant, and t
is time measured in seconds.*
*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition,
Prentice-Hall, 1988.)
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Example 5 APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE (continued)
(a) It is essential for electric generators to rotate at
precisely 60 cycles per second so household
appliances and computers will function properly.
Determine ω for these electric generators.
Each cycle is 2π radians at 60 cycles per second, so
the angular speed is ω = 60(2π) = 120π radians per
second.
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Example 5 APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE (continued)
(b) Graph V in the window [0, .05] by [–200, 200].
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Example 5 APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE (continued)
(c) Determine a value of so that the graph of
is the same as the graph of
Using the negative-angle identity for cosine and a
cofunction identity gives
Therefore, if
Copyright © 2009 Pearson Addison-Wesley 5.4-2
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
5 Trigonometric Identities
Copyright © 2009 Pearson Addison-Wesley 1.1-35.4-3
Sum and Difference Identities
for Sine and Tangent5.4
Sum and Difference Identities for Sine ▪ Sum and Difference
Identities for Tangent ▪ Applying the Sum and Difference Identities
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Sum and Difference Identities for Sine
Cofunction identity
We can use the cosine sum and difference identities
to derive similar identities for sine and tangent.
Cosine difference identity
Cofunction identities
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Sum and Difference Identities for Sine
Sine sum identity
Negative-angle identities
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Sum and Difference Identities for Tangent
Fundamental identity
We can use the cosine sum and difference identities
to derive similar identities for sine and tangent.
Sum identities
Multiply numerator and denominator by 1.
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Sum and Difference Identities for Tangent
Multiply.
Simplify.
Fundamental identity
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Sum and Difference Identities for Tangent
Replace B with –B and use the fact that tan(–B) to
obtain the identity for the tangent of the difference of
two angles.
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Example 1(a) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of sin 75 .
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Example 1(b) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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Example 1(c) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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Example 2 WRITING FUNCTIONS AS EXPRESSIONS
INVOLVING FUNCTIONS OF θ
Write each function as an expression involving
functions of θ.
(a)
(b)
(c)
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Example 3 FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B
Suppose that A and B are angles in standard position
with
Find each of the following.
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Example 3 FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
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 cos A, tan A, sin B and tan B.
Because A is in quadrant II, cos A is negative and
tan A is negative.
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Example 3 FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
Because B is in quadrant III, sin B is negative and
tan B is positive.
Copyright © 2009 Pearson Addison-Wesley 1.1-185.4-18
Example 3 FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
(a)
(b)
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Example 3 FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
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 positive is quadrant I, so
(A + B) is in quadrant IV.
Copyright © 2009 Pearson Addison-Wesley 1.1-205.4-20
Example 4 VERIFYING AN IDENTITY USING SUM
AND DIFFERENCE IDENTITIES
Verify that the equation is an identity.
Copyright © 2009 Pearson Addison-Wesley 5.5-2
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
5 Trigonometric Identities
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Double-Angle Identities5.5Double-Angle Identities ▪ An Application ▪ Product-to-Sum and
Sum-to-Product Identities
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Double-Angle Identities
We can use the cosine sum identity to derive
double-angle identities for cosine.
Cosine sum identity
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Double-Angle Identities
There are two alternate forms of this identity.
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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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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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Example 1 FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ
Given and sin θ < 0, find sin 2θ, cos 2θ, and
tan 2θ.
The identity for sin 2θ requires sin θ.
Any of the three
forms may be used.
Copyright © 2009 Pearson Addison-Wesley 1.1-105.5-10
Example 1 FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Now find tan θ and then use the tangent double-
angle identity.
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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.
Copyright © 2009 Pearson Addison-Wesley 1.1-125.5-12
Example 2 FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ
Find the values of the six trigonometric functions of θ if
Use the identity to find sin θ:
θ is in quadrant II, so sin θ is positive.
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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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Example 3 VERIFYING A DOUBLE-ANGLE IDENTITY
Quotient identity
Verify that is an identity.
Double-angle identity
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Example 4 SIMPLIFYING EXPRESSION DOUBLE-
ANGLE 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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where V is the voltage and R is a constant that
measure the resistance of the toaster in ohms.*
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
*(Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition,
Prentice-Hall, 1988.)
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?
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Example 6 DETERMINING WATTAGE
CONSUMPTION
There are six oscillations.
Copyright © 2009 Pearson Addison-Wesley 5.5-19
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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Product-to-Sum Identities
Similarly, subtracting cos(A + B) from cos(A – B)
gives a product-to-sum identity for sines.
Copyright © 2009 Pearson Addison-Wesley 5.5-21
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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Example 7
Write 4 cos 75° sin 25° as the sum or difference of
two functions.
USING A PRODUCT-TO-SUM IDENTITY
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Example 8
Write as a product of two functions.
USING A SUM-TO-PRODUCT IDENTITY
Copyright © 2009 Pearson Addison-Wesley 5.6-2
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
5 Trigonometric Identities
Copyright © 2009 Pearson Addison-Wesley 1.1-35.6-3
Half-Angle Identities5.6Half-Angle Identities ▪ Applying the Half-Angle Identities
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Half-Angle Identities
We can use the cosine sum identities to derive half-
angle identities.
Choose the appropriate sign depending on the
quadrant of
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Half-Angle Identities
Choose the appropriate sign depending on the
quadrant of
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Half-Angle Identities
There are three alternative forms for
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Example 1 USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of cos 15 using the half-angle
identity for cosine.
Choose the positive square root.
Copyright © 2009 Pearson Addison-Wesley 1.1-105.6-10
Example 2 USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of tan 22.5 using the identity
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Example 3 FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s
The angle associated with lies in quadrant II since
is positive while are negative.
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Example 3 FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s (cont.)
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Example 4 SIMPLIFYING EXPRESSIONS USING
THE HALF-ANGLE IDENTITIES
Simplify each expression.
Substitute 12x for A:
This matches part of the identity for
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