1 Equivalence
To verify that an equation in
θ
theta is an identity, show that both of its sides have equal values for each possible replacement for
θ
.
theta .
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Trigonometric Identities (Lesson 14-1)
2
tan
θ
cos
2
θ
=
2
sin
θ
cos
2
θ
cos
θ
=
2
sin
θ
cos
θ
table with 2 rows and 2 columns , row1 column 1 , 2 tangent theta , cosine squared , theta , column 2 equals . fraction 2 sine theta , cosine squared , theta , over cosine theta end fraction , row2 column 1 , , column 2 equals 2 sine theta cosine theta , end table
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Angle Identities, and Double- and Half-Angle Identities. (Lessons 14-6 and 14-7)
sin
(
−
θ
)
=
−
sin
θ
sin
(
π
2
−
θ
)
=
cos
θ
sin
(
A
+
B
)
=
sin
A
cos
B
−
sin
B
cos
A
sin
2
θ
=
2
sin
θ
cos
θ
sin
A
2
=
±
1
−
cos
A
2
table with 5 rows and 1 column , row1 column 1 , sine . open , negative theta , close . equals negative sine theta , row2 column 1 , sine . open . pi over 2 , minus theta . close . equals cosine theta , row3 column 1 , sine . open , eh plus b , close . equals sine eh cosine b minus sine b cosine eh , row4 column 1 , sine 2 theta equals 2 sine theta cosine theta , row5 column 1 , sine , eh over 2 , equals plus minus . square root of fraction 1 minus cosine eh , over 2 end fraction end root , end table
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2 Function
If the domain of a trigonometric function is appropriately restricted, its inverse is a function. |
Inverse Trigonometric Functions (Lesson 14-2)
y
=
sin
θ
0
≤
θ
≤
π
θ
=
sin
−
1
x
y
=
cos
θ
−
π
2
≤
θ
≤
π
2
θ
=
cos
−
1
x
y
=
tan
θ
−
π
2
<
θ
<
π
2
θ
=
tan
−
1
x
table with 3 rows and 3 columns , row1 column 1 , y equals sine theta , column 2 0 less than or equal to theta less than or equal to pi , column 3 theta equals . sine super negative 1 end super . x , row2 column 1 , y equals cosine theta , column 2 negative , pi over 2 , less than or equal to theta less than or equal to , pi over 2 , column 3 theta equals . cosine super negative 1 end super . x , row3 column 1 , y equals tangent theta , column 2 negative , pi over 2 , less than theta less than , pi over 2 , column 3 theta equals . tangent super negative 1 end super . x , end table
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3 Equivalence
The trigonometric function values of
θ
,
0
°
<
θ
<
90
°
,
theta comma 0 degrees less than theta less than 90 degrees comma are the trigonometric ratios for a right triangle. |
Right Triangles and Trigonometric Ratios (Lessons 14-3)
sin
θ
=
OPP
HYP
=
y
r
cos
θ
=
ADJ
HYP
=
x
r
tan
θ
=
OPP
ADJ
=
y
x
table with 3 rows and 2 columns , row1 column 1 , sine theta , column 2 equals , fraction
Image Long Description
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Area, the Law of Sines, and the Law of Cosines (Lessons 14-4 and 14-5)
Area
Δ
A
B
C
:
1
2
b
c
sin
A
Law
of
Sines
:
sin
A
a
=
sin
B
b
=
sin
C
c
Law
of
Cousines
:
a
2
=
b
2
+
c
2
−
2
b
c
cos
A
table with 3 rows and 2 columns , row1 column 1 , cap area , cap delta eh b c colon , column 2 1 half , b c sine eh , row2 column 1 , cap law of , cap sines , colon , column 2 fraction sine eh , over eh end fraction . equals . fraction sine b , over b end fraction . equals . fraction sine c , over c end fraction , row3 column 1 , cap law of , cap cousines , colon , column 2 eh squared , equals , b squared , plus , c squared , minus 2 b c cosine eh , end table
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