Download CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS
Shiah-Sen Wang
The graphs are prepared by Chien-Lun Lai
Based on : Precalculus: Mathematics for Calculus
by J. Stuwart, L. Redin & S. Watson,
6th edition, 2012, Brooks/Cole
Chapter 5-7.
LECTURE 6. TRIGONOMETRY: PART I
This lecture is the first part of reviewing high school
trigonometry: Pythagorean theorem, the definitions of
trigonometric and inverse trigonometric functions and some of
their properties.
Trigonometry of Right Triangle
Angle Measure: If a circle of radius 1 (so its circumference is
2π) with vertex of an angle at its center, then the measure of
this angle in radians (abbreviated rad) is the length of the arc
that subtends the angle. See
where the second graph shows that the length s of an arc that
subtends a central angle of θ radians in a circle of radius r is
s = rθ
Trigonometry of Right Triangle
RELATIONSHIP BETWEEN DEGREES AND RADIANS
180○ = π rad,
1 rad = (
180 ○
)
π
1○ =
π
rad
180
Examples
1. Express 30○ in radians.
2. Express 45○ in radians.
3. Express
π
rad in degrees.
3
π
π
) rad = rad.
180
6
π
π
○
45 = 45 (
) rad = rad.
180
4
π
π 180
rad = ( ) (
) = 60○ .
3
3
π
30○ = 30 (
A note on terminology: We often use a phrase such as “a 30○
degree angle” to mean an angle whose measure is 30○ . Also
for an angle θ, we write θ = 30○ or θ = π/6 to mean the measure
of θ is 30○ or π/6rad. When no unit is given, the angle is
assume to be in radians.
Area of a Circular Sector
In a disc of radius r , the area of a sector with central angle of θ
radians
is
A = πr 2 ×
θ
1
= πr 2 .
2π 2
Trigonometry of Right Triangle
Pythagorean Theorem: In any right triangle, the square of the
length of the hypotenuse equals the sum of the squares of the
lengths of the two other sides. If the hypotenuse has length c,
and the legs have lengths a and b, then
c 2 = a2 + b 2 .
Trigonometric Function: Right Triangle Approach
Trigonometric Ratios for 0 < θ <
b
c
c
csc θ =
b
sin θ =
a
c
c
sec θ =
a
cos θ =
π
2
b
a
a
cot θ =
b
tan θ =
Examples
π
1
and cos θ = . Find the values of
2
3
sin θ, tan θ, cot θ, sec θ and csc θ.
By
1. Suppose 0 < θ <
we
length of the opposite side is
¿ see that the √
√
2
Á
√
2
1
2 2
Á
À
1−( ) =
= sin θ, tan θ =
, cot θ = 2 2,
3√
3
4
3 2
csc θ =
and sec θ = 3
4
Examples
π
and tan θ = 2. Find the values of
2
sin θ, cos θ, cot θ, sec θ and csc θ.
Use
2. Suppose 0 < θ <
√
We see the length of the hypotenuse side is 5. Hence
√
2
1
1
sin θ = √ , cos θ = √ , cot θ = , sec θ = 5 and csc θ =
2
5
5
√
5
2
Trigonometric Function: Unit Circle Approach
The unit circle is the circle of radius 1 centered at the origin in
the xy-plane. Its equation is
x 2 + y 2 = 1.
Suppose θ is a real number. Mark off a distance θ along the
unit circle, starting at the point (1, 0) and moving in a
counterclockwise direction if θ > 0 or in a clockwise direction if
θ < 0. In this way we arrive at a point P(x, y ) on the unit circle.
The point P(x, y ) obtained in this way is called the terminal
point determined by the real number θ.
Trigonometric Function: Unit Circle Approach
,
Trigonometric Function: Unit Circle Approach
Trigonometric Functions for θ ∈ R
sin θ = y
csc θ =
1
(y ≠ 0)
y
cos θ = x
sec θ =
1
(x ≠ 0)
x
tan θ =
y
(x ≠ 0)
x
cot θ =
x
(y ≠ 0)
y
π
, a = x, b = y and c = 1, the unit circle
2
approach is identical with the right triangle approach in the
defining these trigonometric functions, but the domains of these
trigonometric functions are larger using the unit circle approach.
Note that, when 0 < θ <
Trigonometric Function: Special Triangles
Special Triangles
θ in
0○
30○
45○
60○
90○
○
θ in rad
0
π
6
π
4
π
3
π
2
sin θ
0
1
√2
2
2
√
3
2
1
cos θ
1
√
3
√2
2
2
1
2
0
tan θ
0
√
3
3
csc θ
−
2
√
1
√
3
−
2
√
2 3
3
1
sec θ
1
√
2 3
3
√
2
2
−
cot θ
−
√
3
1
√
3
3
0
Trigonometric Function: Special Triangles
Fundamental Identities of Trigonometric Function
Reciprocal Identities
csc θ =
1
sin θ
sec θ =
1
cos θ
cot θ =
1
tan θ
sin θ
cos θ
cot θ =
cos θ
sin θ
Pythagorean Identities
tan θ =
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
Fundamental Identities of Trigonometric Function
Even-Odd Identities
sin(−θ) = − sin θ
cos(−θ) = cos θ
tan(−θ) = − tan θ
csc(−θ) = − csc θ
sec(−θ) = sec θ
cot(−θ) = − cot θ
Fundamental Identities of Trigonometric Function
Cofunction Identities
sin (
π
− θ) = cos θ
2
tan (
π
− θ) = cot θ
2
sec (
π
− θ) = csc θ
2
π
− θ) = sin θ
2
cot (
π
− θ) = tan θ
2
csc (
π
− θ) = secθ
2
cos (
Examples on Applications of Trigonometric Identities
cos θ
.
1 + sin θ
cos θ
sin θ
cos θ
sin θ(1 + sin θ) + cos2 θ
tan θ +
=
+
=
1 + sin θ cos θ 1 + sin θ
cos θ(1 + sin θ)
sin θ + sin2 θ + cos2 θ
1 + sin θ
1
=
=
=
= sec θ
cos θ(1 + sin θ)
cos θ(1 + sin θ) cos θ
1 − sin θ
2. Prove
= (sec θ − tan θ)2 .
1 + sin θ
1 − sin θ 1 − sin θ 1 − sin θ (1 − sin θ)2
1 − sin θ 2
=
⋅
=
=
(
)
1 + sin θ 1 + sin θ 1 − sin θ
cos θ
1 − sin2 θ
1
sin θ 2
=(
−
) = (sec θ − tan θ)2 .
cos θ cos θ
1. Simplify tan θ +
Trigonometric Graphs
A function f is periodic if there is a positive p such that
f (θ + p) = f (θ) for every θ. The least such positive number is the
period of f . If f has period p, then the graph of f on any interval
of length p is called one complete period of f .
Periodic Properties of Sine and Cosine
The functions sine and cosine have period 2π:
sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
Trigonometric Graphs
Periodic Properties of tangent and Cotangent
The functions tangent and cotangent have period π:
tan(θ + π) = tan θ
cot(θ + π) = cot θ
Trigonometric Graphs
Periodic Properties of Secant and Cosecant
The functions secant and cosecant have period 2π:
sec(θ + 2π) = sec θ
csc(θ + 2π) = csc θ
Examples
√
2π
π π
π
π
3
1. sin
= sin ( + ) = cos (− ) = cos =
.
3
2 6
6
6
2
π π
π
π
2. cos π = cos ( + ) = sin (− ) = − sin = −1.
2 2
2
2
5π
π
π π
π π
3. cot
= cot [ + ( + )] = tan [− ( + )]
4
2
2 4
2 4
= − tan ( π2 + π4 ) = − cot (− π4 ) = cot π4 = 1.
π
π
π
3π
= tan (2π − ) = tan (− ) = − tan = 0.
4. tan
2
2
2
2
√
5π
π
π
π 2 3
5. sec
= sec (2π − ) = sec (− ) = sec =
.
6
6
6
6
3
31π
π
π
π π
6. csc
= csc (5π + ) = csc (π + ) = sec [− ( + )]
6
6
6
2 6
= sec ( π2 + π6 ) = csc (− π6 ) = − csc π6 = − sin1 π = − 11 = −2.
6
2
Domains and Ranges of Trigonometric Functions
sin ∶ R Ð→ [−1, 1]
cos ∶ R Ð→ [−1, 1]
tan ∶ R ∖ {k π +
π
∣ k ∈ Z} Ð→ R
2
cot ∶ R ∖ {k π ∣ k ∈ Z} Ð→ R
sec ∶ R ∖ {kπ ∣ k ∈ Z} Ð→ (−∞, −1] ∪ [1, ∞)
csc ∶ R ∖ {k π +
π
∣ k ∈ Z} Ð→ (−∞, −1] ∪ [1, ∞)
2
Signs of Trigonometric Functions
Quadrant
I
II
III
IV
Positive Functions
all
sin, csc
tan, cot
cos, sec
Negative Functions
none
cos, sec, tan, cot
sin, csc, cos, sec
sin, csc, tan, cot
Inverse Trigonometric Functions
● The Inverse Sine Function is the function
π π
sin−1 ∶ [−1, 1] Ð→ [− , ]
2 2
defined by
sin−1 x = y ⇐⇒ sin y = x
The inverse sine function is also called arcsine, denoted
by arcsin.
Inverse Trigonometric Functions
Examples
1 π
= .
2 √
6
π
3
arcsin (−
)=− .
2
3
−1
sin 2 =? Because 2 > 1, 2 is not in the domain of arcsin,
so sin−1 2 is not defined.
π
π
arcsin (sin ) = .
4
4
1
π
7π
−1
) = sin−1 (− ) = − .
sin (sin
6
2
6
i sin−1
ii
iii
iv
v
Inverse Trigonometric Functions
● The Inverse Cosine Function is the function
cos−1 ∶ [−1, 1] Ð→ [0, π]
defined by
cos−1 x = y ⇐⇒ cos y = x
The inverse cosine function is also called arccosine,
denoted by arccos.
Inverse Trigonometric Functions
● The Inverse Tangent Function is the function
π π
tan−1 ∶ R Ð→ (− , )
2 2
defined by
tan−1 x = y ⇐⇒ tan y = x
The inverse tangent function is also called arctangent,
denoted by arctan.
Inverse Trigonometric Functions
● The Inverse Cotangent Function is the function
cot−1 ∶ R Ð→ (0, π)
defined by
cot−1 x = y ⇐⇒ cot y = x
The inverse cotangent function is also called
arccotangent, denoted by arccot.
Inverse Trigonometric Functions
● The Inverse Secant Function is the function
π
π
sec−1 ∶ {x ∈ R ∣ ∣x∣ ≥ 1} Ð→ [0, ) ∪ ( , π]
2
2
defined by
sec−1 x = y ⇐⇒ sec y = x
The inverse cosecant function is also called arcsecant,
denoted by arcsec.
Inverse Trigonometric Functions
● The Inverse Cosecant Function is the function
π
π
csc−1 ∶ {x ∈ R ∣ ∣x∣ ≥ 1} Ð→ [− , 0) ∪ (0, ]
2
2
defined by
csc−1 x = y ⇐⇒ csc y = x
The inverse cosecant function is also called arccosecant,
denoted by arccsc.
Inverse Trigonometric Functions
Examples
2π
1 π
= sin−1 = .
3
2 6
π
−1
2. cos tan (− ) = cos−1 (−1) = π.
4
√
1
π
3
3. tan arcsin = tan =
.
2
6
3
√
π √
2
== csc = 2.
4. csc cos−1
2
4
1. sin−1 cos