7
ANALYTIC TRIGONOMETRY
104
Since 22.5 is in the rst quadrant, we choose the + sign.
45
sin 22.5 = sin
2
r
1 cos 45
=
2
s
p
2
1
2
=
2
s
p
2
2
=
4
Example: Find tan
u
2
if sin u =
2
5
and u is in quadrant II.
solution: From the formula,
tan
u
2
=
1
cos u
s
7
Analytic Trigonometry
7.1
Trigonometric Identities
Lets begin by listing the identities we already know.
Reciprocal Identities:
1
sin
sec =
1
cos
cot =
tan =
csc =
sin
cos
cot =
1
tan
cos
sin
Pythagorean Identities:
sin2 + cos2 = 1
tan2 + 1 = sec
7
ANALYTIC TRIGONOMETRY
113
5
31
k<
2.58
12
12
The ks in this range are k = 0, k = 1 and k = 2. Hence
x=
3. Consider the equation
p
3 tan( x )
2
5 17 29
,
,
18 18 18
1 = 0.
(a) Find all the solutions of the equation.
(b) Find all the solutions in the int
7
ANALYTIC TRIGONOMETRY
7.5
110
Trigonometric Equations
One frequently has to solve equations involving trig functions. Sometimes the values of x
you look at are restricted, while others you are asked to nd all the values of x that make
a given equation t
7
ANALYTIC TRIGONOMETRY
107
(c) We know sin x is never 3 (it is never greater than 1), so sin 1 ( 3 ) is undened.
2
2
p
(d) For u 2
, , we have cos(u) = 1 sin2 u, so
2 2
s
3
3
1
2
1
cos sin
=
1 sin sin
5
5
s
2
3
=
1
5
r
9
=
1
25
r
16
=
25
4
=
5
Inv
7
ANALYTIC TRIGONOMETRY
101
RHS =
=
=
tan2
sec 1
sin2
cos2
1
cos
sin2
cos2
1 cos
cos
2
1
sin
cos
2 1
cos
cos
2
(1 cos ) cos
=
cos2 (1 cos )
(1 cos )(1 + cos )
=
cos (1 cos )
1 + cos
=
= RHS
cos
=
7.2
Addition and Subtraction Formulas
Formula
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