Transcript Chapter 5 Section 3

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
Copyright © 2009 Pearson Addison-Wesley
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
Copyright © 2009 Pearson Addison-Wesley
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)).
Copyright © 2009 Pearson Addison-Wesley
5.2-4
Difference Identity for Cosine
Since the central angles
SOQ and POR are
equal, PR = SQ.
Using the distance formula,
we have
Copyright © 2009 Pearson Addison-Wesley
5.2-5
Difference Identity for Cosine
Square both sides and clear parentheses:
Rearrange the terms:
Copyright © 2009 Pearson Addison-Wesley
5.2-6
Difference Identity for Cosine
Subtract 2, then divide by –2:
Copyright © 2009 Pearson Addison-Wesley
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
Copyright © 2009 Pearson Addison-Wesley
5.2-8
Cosine of a Sum or Difference
Copyright © 2009 Pearson Addison-Wesley
1.1-9
5.2-9
Example 1(a) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of cos 15.
Copyright © 2009 Pearson Addison-Wesley
1.1-10
5.2-10
Example 1(b) FINDING EXACT COSINE FUNCTION
VALUES
Find the exact value of
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-12
5.2-12
Cofunction Identities
Similar identities can be obtained for a
real number domain by replacing 90
with
Copyright © 2009 Pearson Addison-Wesley
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°)
Copyright © 2009 Pearson Addison-Wesley
1.1-14
5.2-14
Example 2
USING COFUNCTION IDENTITIES TO
FIND θ
Find an angle that satisfies each of the following:
(c)
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
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
Copyright © 2009 Pearson Addison-Wesley
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
Copyright © 2009 Pearson Addison-Wesley
1.1-20
5.2-20
Example 4
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
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.)
Copyright © 2009 Pearson Addison-Wesley
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.
Copyright © 2009 Pearson Addison-Wesley
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].
Copyright © 2009 Pearson Addison-Wesley
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
Copyright © 2009 Pearson Addison-Wesley
1.1-26
5.2-26