5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
5.1-1
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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5.1-2
5.1 Fundamental Identities
Fundamental Identities Using the Fundamental Identities
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1.1-3
5.1-3
Fundamental Identities
Reciprocal Identities
Quotient Identities
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1.1-4
5.1-4
Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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1.1-5
5.1-5
Note
In trigonometric identities, θ can be
an angle in degrees, an angle in
radians, a real number, or a variable.
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1.1-6
5.1-6
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT
Example 1
If
and θ is in quadrant II, find each function
value.
(a) sec θ
Pythagorean
identity
In quadrant II, sec θ is negative, so
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1.1-7
5.1-7
Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
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1.1-8
5.1-8
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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1.1-9
5.1-9
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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1.1-10
5.1-10
Example 2
EXPRESSING ONE FUNCITON IN
TERMS OF ANOTHER
Express cos x in terms of tan x.
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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1.1-11
5.1-11
Example 3
REWRITING AN EXPRESSION IN
TERMS OF SINE AND COSINE
Write tan θ + cot θ in terms of sin θ and cos θ, and
then simplify the expression.
Quotient identities
Write each fraction
with the LCD.
Pythagorean identity
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1.1-12
5.1-12
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.
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1.1-13
5.1-13
5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
5.2-1
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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5.2-2
5.2 Verifying Trigonometric
Identities
Verifying Identities by Working With One Side ▪ Verifying
Identities by Working With Both Sides
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1.1-3
5.2-3
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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1.1-4
5.2-4
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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1.1-5
5.2-5
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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1.1-6
5.2-6
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.
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1.1-7
5.2-7
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.
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1.1-8
5.2-8
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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1.1-9
5.2-9
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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5.2-10
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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1.1-11
5.2-11
Example 2
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that
is an identity.
Left side
Distributive
property
Right side
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1.1-12
5.2-12
Example 3
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that
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is an identity.
1.1-13
5.2-13
Example 4
VERIFYING AN IDENTITY (WORKING
WITH ONE SIDE)
Verify that
is an identity.
Multiply by 1
in the form
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1.1-14
5.2-14
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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5.2-15
Example 5
VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES)
Verify that
identity.
is an
Working with the left side:
Multiply by 1
in the form
Distributive
property
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1.1-16
5.2-16
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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1.1-17
5.2-17
VERIFYING AN IDENTITY (WORKING
WITH BOTH SIDES) (continued)
Example 5
Right side of given
equation
Left side of given
equation
Common third
expression
So, the identity is verified.
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1.1-18
5.2-18
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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1.1-19
5.2-19
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.)
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1.1-20
5.2-20
Example 6
APPLYING A PYTHAGOREAN IDENTITY
TO RADIOS (continued)
Factor.
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1.1-21
5.2-21
5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
5.2-1
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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5.2-2
5.3 Sum and Difference
Identitites for Cosine
Difference Identity for Cosine ▪ Sum Identity for Cosine ▪
Cofunction Identities ▪ Applying the Sum and Difference Identities
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1.1-3
5.2-3
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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5.2-4
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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5.2-5
Difference Identity for Cosine
Square both sides and clear parentheses:
Rearrange the terms:
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5.2-6
Difference Identity for Cosine
Subtract 2, then divide by –2:
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5.2-7
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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5.2-8
Cosine of a Sum or Difference
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1.1-9
5.2-9
Example 1(a) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of cos 15 .
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1.1-10
5.2-10
Example 1(b) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of
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1.1-11
5.2-11
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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1.1-12
5.2-12
Cofunction Identities
Similar identities can be obtained for a
real number domain by replacing 90
with
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1.1-13
5.2-13
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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1.1-14
5.2-14
Example 2
USING COFUNCTION IDENTITIES TO
FIND θ
Find an angle that satisfies each of the following:
(c)
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1.1-15
5.2-15
Note
Because trigonometric (circular)
functions are periodic, the solutions
in Example 2 are not unique. Only
one of infinitely many possiblities
are given.
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1.1-16
5.2-16
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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5.2-17
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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1.1-18
5.2-18
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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1.1-19
5.2-19
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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1.1-20
5.2-20
Example 4
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FINDING cos (s + t) GIVEN
INFORMATION ABOUT s AND t (cont.)
1.1-21
5.2-21
Note
The values of cos s and sin t could
also be found by using the
Pythagorean identities.
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1.1-22
5.2-22
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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1.1-23
5.2-23
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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1.1-24
5.2-24
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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1.1-25
5.2-25
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
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1.1-26
5.2-26
5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
5.4-1
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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5.4-2
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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1.1-3
5.4-3
Sum and Difference Identities
for Sine
We can use the cosine sum and difference identities
to derive similar identities for sine and tangent.
Cofunction identity
Cosine difference identity
Cofunction identities
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5.4-4
Sum and Difference Identities
for Sine
Sine sum identity
Negative-angle identities
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5.4-5
Sine of a Sum or Difference
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1.1-6
5.4-6
Sum and Difference Identities
for Tangent
We can use the cosine sum and difference identities
to derive similar identities for sine and tangent.
Fundamental identity
Sum identities
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Multiply numerator and
denominator by 1.
5.4-7
Sum and Difference Identities
for Tangent
Multiply.
Simplify.
Fundamental
identity
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5.4-8
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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5.4-9
Tangent of a Sum or Difference
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1.1-10
5.4-10
Example 1(a) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of sin 75 .
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1.1-11
5.4-11
Example 1(b) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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1.1-12
5.4-12
Example 1(c) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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1.1-13
5.4-13
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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1.1-14
5.4-14
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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1.1-15
5.4-15
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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1.1-16
5.4-16
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.
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1.1-17
5.4-17
Example 3
FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
(a)
(b)
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1.1-18
5.4-18
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.
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1.1-19
5.4-19
Example 4
VERIFYING AN IDENTITY USING SUM
AND DIFFERENCE IDENTITIES
Verify that the equation is an identity.
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1.1-20
5.4-20
5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
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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1.1-3
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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1.1-9
5.5-9
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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1.1-11
5.5-11
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
5.5-12
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
5.5-13
Example 3
VERIFYING A DOUBLE-ANGLE IDENTITY
Verify that
is an identity.
Quotient identity
Double-angle
identity
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1.1-14
5.5-14
Example 4
SIMPLIFYING EXPRESSION DOUBLEANGLE IDENTITIES
Simplify each expression.
Multiply by 1.
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1.1-15
5.5-15
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
5.5-16
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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1.1-17
5.5-17
Example 6
DETERMINING WATTAGE
CONSUMPTION
There are six oscillations.
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1.1-18
5.5-18
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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1.1-22
5.5-22
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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1.1-23
5.5-23
Sum-to-Product Identities
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1.1-24
5.5-24
Example 8
Write
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USING A SUM-TO-PRODUCT IDENTITY
as a product of two functions.
1.1-25
5.5-25
5
Trigonometric
Identities
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5.6-1
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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5.6-2
5.6 Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities
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1.1-3
5.6-3
Half-Angle Identities
We can use the cosine sum identities to derive halfangle identities.
Choose the appropriate sign depending on the
quadrant of
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5.6-4
Half-Angle Identities
Choose the appropriate sign depending on the
quadrant of
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5.6-5
Half-Angle Identities
There are three alternative forms for
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5.6-6
Half-Angle Identities
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5.6-7
Double-Angle Identities
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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.
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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
is positive while
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lies in quadrant II since
are negative.
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Example 3
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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.
This matches part of the identity for
Substitute 12x for A:
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Example 5
VERIFYING AN IDENTITY
Verify that
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is an identity.
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