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Chapter 8. Analytic Trigonometry
8.1 Trigonometric Identities
Fundamental Identities
Reciprocal Identities:
csc x =
1
sin x
sec x =
1
cos x
cot x =
1
tan x
tan x =
1
cot x
tan x =
sin x
cos x
cot x =
cos x
sin x
Pythagorean Identities:
cos2 x + sin2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
Even-Odd Identities:
cos(−x) = cos x,
sec(−x) = sec(x),
sin(−x) = −sin x,
tan(−x) = −tan(x),
csc(−x) = −csc x,
cot(−x) = −cot(x).
Cofunction Identities:
π

sin  − u  = cos u
2

π

tan  − u  = cot u
2

π

sec − u  = csc u
2

π

cos − u  = sin u
2

π

cot  − u  = tan u
2

π

csc − u  = sec u
2

Proving Trigonometric Identities
Example: [#46]
sin4 x − cos4 x = sin2 x − cos2 x
Example: [#61]
sec x
= sec x(sec x + tan x)
sec x − tan x
Example: [#53]
tan2 x − sin2 x = tan2 x sin2 x
Example: [#64]
sin x
− cot x = csc x
1 − cos x
Example: [#79]
(tan x + cot x)2 = sec2 x + csc2 x
8.2 Addition and Subtraction Formulas
Addition / Subtraction formulas
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
tan(α + β ) =
tan α + tan β
1 − tan α tan β
tan(α − β ) =
tan α − tan β
1 + tan α tan β
Example: Evaluate sin θ, cos θ, and tan θ, for θ = 5π / 12.
Example: Evaluate sin(80º) cos(20º) − cos(80º) sin(20º)
8.3 Double-Angle, Half-Angle, and Product-Sum Formulas
Double-angle formulas
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2sin2 x
tan 2 x =
2 tan x
1 − tan 2 x
Half-angle formulas (simple versions, a.k.a. Power Reducing Formulas)
sin2 x = (1 − cos 2x ) / 2
cos2 x = (1 + cos 2x ) / 2
tan 2 x =
1 − cos 2 x
1 + cos 2 x
Half-angle formulas
sin
u
1 − cos u
=±
2
2
cos
u
1 + cos u
=±
2
2
(The sign is determined by the quadrant in which u / 2 lies.)
tan
u 1 − cos u
sin u
=
=
2
sin u
1 + cos u
Product-to-Sum Formulas
sin(α) sin(β) = [cos(α − β) − cos(α + β)] / 2
sin(α) cos(β) = [sin(α − β) + sin(α + β)] / 2
cos(α) cos(β) = [cos(α − β) + cos(α + β)] / 2
Sum-to-Product Formulas
sin α + sin β = 2 sin
sin α − sin β = 2 cos
α+β
2
cos
α +β
cos α + cos β = 2 cos
2
α+β
cos α − cos β = −2 sin
2
sin
2
α −β
cos
α +β
2
α −β
2
α −β
sin
2
α −β
2
8.4 Inverse Trigonometric Functions
While trigonometric functions are not one-to-one in their respective nature
domain (by definition, no periodic function, in general, is one-to-one),
consequently they don’t have inverse functions for their “native” domains.
However, if we restrict their domains (to very small intervals) then it is
possible to define inverse trigonometric functions.
The Inverse Sine Function
The inverse sine function, sin −1, has domain [−1, 1] and range [−π / 2,
π / 2]. It is defined by
sin −1 x = y
sin y = x
<=>
The inverse sine function is also called arcsine, denoted by arcsin.
sin(sin −1 x) = x
for
−1 ≤ x ≤ 1
sin −1(sin x) = x
for
− π/2 ≤ x ≤ π/2
Graph of y = arcsin x
The Inverse Cosine Function
The inverse cosine function, cos −1, has domain [−1, 1] and range [0, π].
It is defined by
cos
−1
x=y
cos y = x
<=>
The inverse cosine function is also called arccosine, denoted by arccos.
cos(cos
−1
x) = x
for
−1 ≤ x ≤ 1
cos (cos x) = x
for
0≤x≤π
−1
Graph of y = arccos x
Example: cos(cos
−1
1) = 1, but
−1
cos (cos 5π / 4) = 3π / 4 (Why?)
The Inverse Tangent Function
The inverse tangent function, tan −1, has domain (−∞, ∞) and range (−π / 2,
π / 2). It is defined by
tan −1 x = y
tan y = x
<=>
The inverse tangent function is also called arctangent, denoted by arctan.
tan(tan −1 x) = x
for
all real numbers
tan −1(tan x) = x
for
− π/2 < x < π/2
Graph of y = arctan x
Note the twin horizontal asymptotes y = ± π / 2. It is the first
example we have seen of a function with 2 horizontal asymptotes.
The Inverse Secant Function
The inverse secant function, sec −1, has domain | x | ≥ 1 and range [0, π / 2)
and (π / 2, π]. It is defined by
sec −1 x = y
sec y = x
<=>
The inverse secant function is also called arcsecant, denoted by arcsec.
sec(sec −1 x) = x
for
all x, | x | ≥ 1
sce −1(sec x) = x
for
0 ≤ x < π / 2, or
π/2 < x ≤ π
−1
−1
Note: The choice of intervals for the range of sec , and as well for csc ,
is not universally agreed upon. Your textbook, for example, use [0, π / 2)
−1
and [π , 3π / 2) as the range of sec .
Inverse cotangent and cosecant functions are defined similarly, although
they are not used nearly as commonly as the others.
Graph of y = arcsec x
There is a horizontal asymptote y = π / 2.
Example: Evaluate
(a) cos −1(−1) = π
(b) tan −1(1) = π / 4
(c) sin(cos −1(1 / 2)) =
(d) cos(tan −1(2 / 5)) =
(e) sec(sin −1(1 / 3)) =
(f) cos(cot −1(−4 / 7)) =
(g) csc(tan −1(−2 / 3)) =
3 /2
8.5 Trigonometric Equations
Example: 2sin x + 1 + 0
2
Example: 4cos x = 3
Example: 4sin(3x) cos(3x) = 1
2
Example: 2cos x + 7cos x − 4 = 0
3
2
Example: 2cos x − 3 − 2cos x = −3sin x
Example: [#56]
tan4 x − 13tan2 x + 36 = 0
Example: [#22]
3tan3 x = tan x
Example: [#34]
sec
x
x
= cos
2
2
Find all solution of the given equation on [0, 2π).
Example: [#61] cos x cos 3x − sin x sin 3x = 0
Example: [#65]
sin 2x + cos x = 0
Example: [#54]
2sin 2x − cos x = 0