Transcript lat04_0703

7
Trigonometric
Identities and
Equations
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7.3-1
Trigonometric Identities and
7 Equations
7.1 Fundamental Identities
7.2 Verifying Trigonometric Identities
7.3 Sum and Difference Identities
7.4 Double-Angle and Half-Angle Identities
7.5 Inverse Circular Functions
7.6 Trigonometric Equations
7.7 Equations Involving Inverse
Trigonometric Functions
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7.3 Sum and Difference Identities
Cosine Sum and Difference Identities ▪ Cofunction Identities ▪
Sine and Tangent 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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Cosine of a Sum or Difference
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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 90
with
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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.
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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(180° – θ) as a trigonometric function of θ
alone.
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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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Sum and Difference Identities
for Sine
Sine sum identity
Negative-angle identities
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Sine of a Sum or Difference
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Sum and Difference Identities
for Tangent
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
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denominator by 1.
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) = –tan B to obtain the identity for the
tangent of the difference of two angles.
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Tangent of a Sum or Difference
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Example 4(a) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of sin 75.
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Example 4(b) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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Example 4(c) FINDING EXACT SINE AND TANGENT
FUNCTION VALUES
Find the exact value of
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Example 5
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 6
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 6
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 6
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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Example 6
FINDING FUNCTION VALUES AND THE
QUADRANT OF A + B (continued)
(a)
(b)
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Example 6
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 I.
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Example 7
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  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 7
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  for these electric generators.
Each cycle is 2 radians at 60 cycles per second, so
the angular speed is  = 60(2) = 120 radians per
second.
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Example 7
APPLYING THE COSINE DIFFERENCE
IDENTITY TO VOLTAGE (continued)
(b) Graph V in the window [0, .05] by [–200, 200].
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Example 7
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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