Young Won Lim12/12/09
● Quadrant Angle Trigonometry● Negative Angle Trigonometry● Reference Angle Trigonometry● Sinusoidal Waves
Trigonometry (3A)
Young Won Lim12/12/09
Copyright (c) 2009 Young W. Lim.
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Quadrant Trigonometry 3 Young Won Lim12/12/09
Triangle Trigonometry
a
c
b
A B
C
All Acute Angles
a
c
b
A
C
B
One Obtuse Angle
Right Triangle
ac
bA B
C
Oblique Triangle
sin = a / c
cos = b / c
tan = a / b
sin = ?
cos = ?
tan = ?
sin = ?
cos = ?
tan = ?
0 ˚ 90 ˚ 90 ˚ 180 ˚0 ˚ 90 ˚
Quadrant Trigonometry 4 Young Won Lim12/12/09
Oblique Triangles Trigonometry
asin A
=bsinB
=csinC
c2 = a2 b2−2ab cosC
a2 = b2 c2−2b c cos A
b2 = c2 a2−2c acosB
The Law of Sines
The Law of Cosines
a
c
b
A B
C
All Acute Angles
a
c
b
A
C
B
One Obtuse Angle
sin = sin 180 ˚ − = sin
cos = cos 180 ˚ − = − cos
tan = tan 180 ˚ − = − tan
Assuming
0 ˚ 90 ˚ , 0 ˚ 180 ˚
Quadrant Trigonometry 5 Young Won Lim12/12/09
Trigonometry in Quadrant Angles (1)
a
c
b
A B
C
c
(-x, y)
-x
y
sin = sin 180 ˚ − = sin
cos = cos 180 ˚ − = − cos
tan = tan 180 ˚ − = − tan
(+x, + y)
x
y
b
− cos = x / b
− tan = y / x
sin = y / b
Quadrant Trigonometry 6 Young Won Lim12/12/09
Trigonometry in Quadrant Angles (2)
a
c
b
A B
C
c
(-x, y)
-x
y
sin = sin 180 ˚ − = sin
cos = cos 180 ˚ − = − cos
tan = tan 180 ˚ − = − tan
a
c
b
A B
C
c
(-x, y)
-x
y
sin = y / b
cos = −x / b
tan = − y / x
Quadrant Trigonometry 7 Young Won Lim12/12/09
Trigonometry in Quadrant Angles (3)
(-x, y)
-x
y
a
c
b
A B
C
c
(-x, y)
-x-x
y
+r
r
r
sin = y / r
cos = −x / r
tan = − y / x
r = x2 y2
Isosceles Triangle
sin = y / b
cos = −x / b
tan = − y / x
Quadrant Trigonometry 8 Young Won Lim12/12/09
Trigonometry in Quadrant Angles (4)
(-x, y)
-x
y
+r
sin = y
cos = −x
tan = − y / x
1 = x2 y2
y
−x
−x , y
Unit Circle
r
r
−1 1
sin = y / r
cos = −x / r
tan = − y / x
r = x2 y2
Quadrant Trigonometry 9 Young Won Lim12/12/09
Negative Angle Trigonometry (1)
y
x 1
x , y
−1
1st Quadrant Angle
sin = y
cos = x
tan = y / x
−
−y
x
1
x , − y
−1
4th Quadrant Angle
sin − = − sin = −y
cos − = cos = x
tan− = − tan = − y /x
0 ̊ < α < 90 ̊ –90 ̊ < –α < 0 ̊
Quadrant Trigonometry 10 Young Won Lim12/12/09
Negative Angle Trigonometry (2)
−
−y
−x
1
−x , − y
−1
3rd Quadrant Angle
sin − = − sin = − y
cos − = − cos = −x
tan − = tan = y /x
sin = sin = y
cos = − cos = −x
tan = − tan = − y / x
y
−x 1
−x , y
−1
2nd Quadrant Angle
90 ̊ < β < 180 ̊ –180 ̊ < –β < –90 ̊
Quadrant Trigonometry 11 Young Won Lim12/12/09
Reference Angle (1)
y
−x 1
−x , y
−1
2nd Quadrant Angle θ
y
x 1
x , y
−1
1st Quadrant Angle θ
sin = y
cos = x
tan = y / x
Reference Angle α
sin = sin = y
cos = − cos = −x
tan = − tan = − y / x
= 180 ˚ −
90 ˚ 180 ˚
Quadrant Trigonometry 12 Young Won Lim12/12/09
Reference Angle (2)
−y
1
−x , − y
−1
3rd Quadrant Angle θ
−y
1
x , − y
−1
4th Quadrant Angle θ
Reference Angle α Reference Angle α
sin = − sin = − y
cos = − cos = −x
tan = tan = y / x
sin = − sin = − y
cos = cos = x
tan = − tan = − y / x
= − 180 ˚ = 360 ˚ −
−x x
270 ˚ 360 ˚180 ˚ 270 ˚
Quadrant Trigonometry 13 Young Won Lim12/12/09
Reference Angle (3)
sin = − sin
cos = − cos
tan = tan
sin = − sin
cos = cos
tan = − tan
sin = sin
cos = cos
tan = tan
sin = sin
cos = − cos
tan = − tan
Reference Angle α
All +only sin +
only tan +
only cos +
= − = 2−
= − =
A Quadrant Angle θ
Quadrant Trigonometry 14 Young Won Lim12/12/09
Making a Helix
Transparent OHP Film
Quadrant Trigonometry 15 Young Won Lim12/12/09
y
z
y
z
x
x
z
y
x
Front View
Top View
Side View
A Helix and Viewpoints
Quadrant Trigonometry 16 Young Won Lim12/12/09
y
z
y
z
y
x
Front View
Side View
0
2
32
Sine Wave
Sine Wave
Quadrant Trigonometry 17 Young Won Lim12/12/09
y
z
y
x
Top View
Side View
0
2
32
y
z
Cosine Wave
Cosine Wave
Quadrant Trigonometry 18 Young Won Lim12/12/09
Symmetry in Sinusoid
Quadrant Trigonometry 19 Young Won Lim12/12/09
Sine and Cosine Waves
2− 312
32
52−
12 0
Sine Wave
Cosine Wave
Quadrant Trigonometry 20 Young Won Lim12/12/09
Sine Wave Symmetry
2− 3
12
32
52−
12
0
Sine Wave
Quadrant Trigonometry 21 Young Won Lim12/12/09
Cosine Wave Symmetry
2− 3
12
32
52−
12
0
Cosine Wave
Young Won Lim12/12/09
References
[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] Blitzer, R. “Algebra & Trigonometry.” 3rd ed, Prentice Hall[4] Smith, R. T., Minton, R. B. “Calculus: Concepts & Connections,” Mc Graw Hill [5] 홍성대, “기본/실력 수학의 정석,”성지출판