Download 1. The Six Trigonometric Functions 1.1 Angles, Degrees

1. The Six Trigonometric Functions
1.1
1.2
1.3
1.4
1.5
Angles, Degrees, and Special Triangles
The Rectangular Coordinate System
Definition I: Trigonometric Functions
Introduction to Identities
More on Identities
1
1.1 Angles, Degrees, and Special Triangles
•
Angles in general
–
–
–
•
Vertex
Initial side, terminal side
Positive angle, negative angle
Angle measurement: degrees
–
•
•
•
Classifying angle by range of angle measurements
Pythagorean theorem
General triangles
Special triangles
–
–
30°- 60°- 90° triangle
45°- 45°- 90° triangle
2
1.1 Angles, Degrees, and Special Triangles
•
Angles in general
–
–
–
Vertex O
Initial side OA, terminal side OB
Positive angle: counter-clockwise; negative angle: clockwise
B
θ
O
A
vertex
3
1.1 Angles, Degrees, and Special Triangles
•
Angles in general
–
–
–
–
–
–
Right angle
Straight angle
Acute angle
Obtuse angle
Complementary angles
Supplementary angles
4
1.1 Angles, Degrees, and Special Triangles
Pythagorean Theorem.
B
2
2
2
a
a +b =c
C
c
b
A
Proof of Pythagorean Theorem.
a
b
b
c
c
a
c
a
c
b
b
a
5
1.1 Angles, Degrees, and Special Triangles
•
General triangles
–
–
–
–
–
–
Equilateral triangle
Isosceles triangle
Scalene triangle
Acute triangle
Obtuse triangle
Right triangle
6
1.1 Angles, Degrees, and Special Triangles
•
Special triangles
–
30°-60°-90° Triangle
60°
2x
x
30°
x 3
–
45°-45°-90° Triangle
45°
x 2
x
45°
x
7
1.1 Angles, Degrees, and Special Triangles
Problems
(1) Indicate whether the given angle is acute or obtuse;
give the complement and supplement [2, 6, 8]
a) 50°
b) 160°
c) y
(2) Refer to figure below [10, 14]
C
αβ
A
ƒ
ƒ
D
B
Find B if β = 45°
Find B if α + β = 80°, and A = 80°
8
1.1 Angles, Degrees, and Special Triangles
Problems
(3) Through how many degrees does the hour hand of a
clock move in 4 hours? [21]
(4) Find the remaining sides of a 30°- 60°- 90° triangle if the
shortest side is 3 [44]
60°
3
(5) Fill the remaining sides of a 45°- 45°- 90° triangle if
the longest side is 12 [58]
9
1.2 The Rectangular Coordinate System
•
•
•
•
•
•
Graphing line
Graphing parabolas
The distance formula
Graphing circles
Angles in standard position
Using technology
10
1.2 The Rectangular Coordinate System
Any point on the y-axis
has the form (0, b)
y-axis
(+, +)
(−, +)
3
Quadrant I
Quadrant II
2
1
−3
−2
Any point on the x-axis
has the form (a, 0)
Origin
x-axis
−1
1
2
3
−1
Quadrant III
(−, −)
−2
−3
Quadrant IV
(+, −)
11
1.2 The Rectangular Coordinate System
(1) Graph line
y = –2x
[8]
(2) Graph parabola y = (x – 1)2 + 2 [Ex]
(3) Find the distance between (–3, 8) and (–1, 6)
[28]
(4) Find the distance between the origin and (x, y) [e.g.4]
(5) Equation of a circle.
•
•
(
)
Verify the point 35 ,− 23 is on the unit circle [40]
Graph circle x2 + y2 = 36 [42]
12
1.2 The Rectangular Coordinate System
90°
60°
120°
135°
45°
150°
30°
(1,0) 0°
360°
180°
330°
210°
315°
225°
240°
300°
270°
Figure 22
13
1.2 The Rectangular Coordinate System
(6) Angle in standard position.
•
•
•
•
An angle in standard position (initial side is on the positive
x-axis, and vertex is the origin)
What does it mean θ ∈ QII (read as angle θ belongs to QII)
Name an angle between 0° and 360° that is co-terminal with
the angle –300° [64]
Draw angle 255° in standard position. Find a positive angle
and a negative angle that are co-terminal with 255° [66]
14
1.2 The Rectangular Coordinate System
(7) Angle in standard position. [70]
•
•
•
•
Draw angle 45° in standard position.
Name a point on the terminal side of the angle
Find the distance from the origin to that point
Name another angle that is co-terminal with 45°
(8) Find all angles that are co-terminal with 150°
[78]
(9) Using technology. Graph a circle.
15
1.3 Definition I: Trigonometric Functions
•
Define six trigonometric functions for angles in
standard position
y
(x, y)
r
y
θ
x
o
x
•
Angle in standard position: initial side is the
positive x-axis and vertex is the origin.
16
1.3 Definition I: Trigonometric Functions
Function
Abbreviation Definition
y
The sine of θ
sinθ
=
r
x
The cosine of θ
cosθ
=
r
y
The tangent of θ
tanθ
=
x
x
The cotangent of θ
cotθ
=
y
r
The secant of θ
secθ
=
x
r
The cosecant of θ
cscθ
=
y
Where x 2 + y 2 = r ,2 or r = x 2 + y 2 . That is, r is the
distance from the origin to (x, y).
17
1.3 Definition I: Trigonometric Functions
For θ in
QI
QII
QIII
QIV
+
+
−
−
cosθ = r and secθ = xr
+
−
−
+
y
tanθ = x and cotθ = xy
+
−
+
−
y
r
sinθ = and cscθ = ry
x
18
1.3 Definition I: Trigonometric Functions
(1) Find six trigonometric functions of θ if (12, –5) is on
the terminal side of θ. [4]
(2) Draw the angle –135° in standard position. Find a point
on the terminal side, then find the sine, cosine, and
tangent of –135°. [26]
(3) Indicate the two quadrants the angle θ could terminate
in if
sinθ = –3/ 10
[42]
(4) Find the remaining trig functions of θ If
•
•
cosθ = 24/25 and θ ∈ QIV
cscθ = 13/5 and cosθ < 0
[52]
[60]
19
1.4 Introduction to Identities
Reciprocal Identities
Equivalent Form
1
sin θ
1
sec θ =
cos θ
sin θ =
1
tan θ
tan θ =
csc θ =
cot θ =
1
csc θ
1
cos θ =
sec θ
1
cot θ
TABLE 1 (MEMORIZE)
20
1.4 Introduction to Identities
Ratio Identities
sin θ
tan θ =
cos θ
sin θ y / r y
=
= = tan θ
Because
cos θ x / r x
cos θ
cot θ =
sin θ
cos θ x / r x
=
= = cot θ
Because
sin θ y / r y
TABLE 2 (MEMORIZE)
21
1.4 Introduction to Identities
Pythagorean Identities
sin 2 θ + cos 2 θ = 1
Equivalent Form
cos θ = ± 1 − sin 2 θ
sin θ = ± 1 − cos 2 θ
1 + tan 2 θ = sec 2 θ
1 + cot 2 θ = csc 2 θ
TABLE 3 (MEMORIZE)
Meaning of sin 2 θ .
sin 2 θ = (sin θ )
2
22
1.4 Introduction to Identities
(1) If cos θ = 23 , find secθ.
[10]
(2) If cotθ = –2, find tanθ. [14]
(3) Find the cotθ, if sin θ =
2
13
and cos θ =
3
13
. [18]
(4) Find the cosθ if sinθ = 23 and θ terminate in QII. [30]
(5) Find secθ if tanθ = 7/24 and cosθ < 0. [42]
(6) Find the rest trigonometric functions of θ, if
•
•
sinθ = 12/13 with θ ∈ QI
[44]
cscθ = 2 and cosθ is negative
[48]
23
1.5 More on Identities
•
•
Extension of 1.4
Use identities in 1.4 to prove more trig identities.
24
1.5 More on Identities
(1) Write cscθ in terms of sinθ only
[7]
(2) Write each expression in terms of sinθ and cosθ, then
simplify
csc θ
(a)
[16]
(b) csc(θ) – cot(θ)cos(θ) [26]
cot θ
(3) Add or subtract, simplify if possible, and write your
answers in sinθ and/or cosθ only [32, 34]
(a) Add.
(b) Subtract.
1
cos θ +
sin θ
1
− cos θ
cos θ
25
1.5 More on Identities
Prove
(4) sinθ cotθ = cosθ
[60]
cos θ
2
(5)
= cos θ [64]
sec θ
26