Chapter 5 Demana Pwrpts

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Chapter 5
Analytic Trigonometry
5.1
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Quick Review
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Identities
functions in calculus.
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Pythagorean Identities
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Cofunction Identities
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Cofunction Identities
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Even-Odd Identities
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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5.2
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Quick Review
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A Proof Strategy
was mathematical proofs are constructed.
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General Strategies I for Proving an Identity
The proof begins with the expression on one side of the identity.
The proof ends with the expression on the other side.
The proof in between consists of showing a sequence of expressions, each one easily seen to be equivalent to its preceding expression.
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General Strategies II for Proving an Identity
Begin with the more complicated expression and work toward the less complicated expression.
If no other move suggests itself, convert the entire expression to one involving sines and cosines.
Combine fractions by combining them over a common denominator.
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General Strategies III for Proving an Identity
Use the algebraic identity (a+b)(a-b) = a2-b2 to set up applications of the Pythagorean identities.
Always be mindful of the “target” expression, and favor manipulations that bring you closer to your goal.
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5.3
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Quick Review
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Cosine of a Difference
Cosine of a Sum
Verifying a Sinusoid Algebraically
These identities provide clear examples of how different the
algebra of functions can be from the algebra of real numbers.
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5.4
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Quick Review
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Double-Angle Identities
Power-Reducing Identities
Half-Angle Identities
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Power-Reducing Identities
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Half-Angle Identities
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5.5
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Quick Review
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Solving Triangles (AAS, ASA)
The Ambiguous Case (SSA)
… and why
The Law of Sines is a powerful extension of the triangle
congruence theorems of Euclidean geometry.
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Example Solving a Triangle Given Two Angles and a Side
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Example Solving a Triangle Given Two Angles and a Side
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Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)
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Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)
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x
15ft
15º
65º
B
A
C
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x
15ft
15º
65º
B
A
C
5.6
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Quick Review
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Solving Triangles (SAS, SSS)
Applications
The Law of Cosines is an important extension of the
Pythagorean theorem, with many applications.
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Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
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Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
-1
-1
22
2
linear factors.
3. 232
4. 961
linear factors.
3. 2
53.130.927 rad
qqqq
sin
tan
cos
sincostan
sin1/103
) that solve tan.
it would make both
x
x
x
xx
xx
xx
xx
xx
xx
xx
x
xxxx
pp
nd cosines only.
.
an identit
h the two expressions
nd cosines only.
.
an identity.
h the two expressions
of special angles (multiples
unique.
holds for t
of special angles (multiples
unique.
function .
sing a calculator.
ne or cosine of an angle:
sincossincos
3443
pppp
3443
sincossincos
3443
sin
34
7
sin
12
uv
pppp
pppp
pp
p
n.
ngle with base length 6
and leg length 4.
n.
ength 6
nctions with no power
): sin2sin.
xx
p
0, , , and .
1.
2.
1.
2.
, , and , respectively,
sinsinsin
ABCABa
sinsinsinsin
sin38sin46sin38sin96
99
9sin469sin96
sin38sin38
10.516
C
ABAC
abac
bc
bc
b
Ð=--=
38 9
46 10.516
96 14.538
ABCabA
==Ð=
o
V
1
sin30sin
76
6sin30
sin
7
30, 7, 25.4, 6, 124.6, 11.5.
AaBbCc
nd a vertical telephone pole
stands beside the road. The angle of el
evation of the Sun is 65, and
the pole casts a 15 foot shadow downhill
along the road. Fin
180906525
651550
sin25sin50
15
15sin50
27.2
sin25
t.
x
BAC
ACB
x
x
a solution to the equation.
1. cos 4/5
2. cos -0.25
or (a) cos and
coefficients that has no real zeros.
xA
a solution to the equation.
1. cos 4/5
2. cos -0.25
or (a) cos and
coefficients that has no
es
2cos
2cos
2cos
ABC
abcbcA
bacacB
cababC
ABCabC
==Ð=
o
V
222
2
2
2cos
161002(4)(10)cos25
6.6
101643.562(4)(6.6)cos
cos0.7659
140
1801402515
14
cababC
c
c
A
A
A
B
A
aBbCc
denote the
Area-.
abcABCs
abcABC
ssasbsc
its.
ss
A

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Chapter 5 Demana Pwrpts