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Young Won Lim1/2/15
● Quadrant Angle Trigonometry● Negative Angle Trigonometry● Reference Angle Trigonometry● Sinusoidal Waves
Trigonometry (3A)
Young Won Lim1/2/15
Copyright (c) 2009 - 2014 Young W. Lim.
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Trigonometry (3A) 3 Young Won Lim1/2/15
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 ˚
Trigonometry (3A) 4 Young Won Lim1/2/15
Oblique Triangles Trigonometry
asin A
=b
sin B=
csinC
c2= a2
b2−2a b cosC
a2= b2
c2−2b c cos A
b2= c2
a2−2c acos B
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
0 ˚ 90 ˚ , 0 ˚ 180 ˚
Trigonometry (3A) 5 Young Won Lim1/2/15
One Obtuse Angle (1)
The Law of Sines
a
c
b
A
C
B
α
sin A =absinB =
abha
=hb
a
c
b
A
C
Bα
h
a
c
b
A
C
B
α
h
sinα =hb
sin A = sinα
asin A
=b
sin B=
csinC
Trigonometry (3A) 6 Young Won Lim1/2/15
One Obtuse Angle (2)
a
c
b
A
C
B
α
a
c
b
C
Bα
hb
A
C
α
h
bcosα
b2= h2
+ b2cos2α
a2= b2
c2−2b c cos A
The Law of Cosines
(bcosα + c)
a2= h2
+ (bcosα + c)2
cos A = (b2 + c2−a2) / 2b c
(b2 + c2−a2) = h2 + b2cos2α + c2
− h2− (bcosα + c)
2
cos A = −2b c cosα / 2b c
cos A = −cosα− h2 − (b cosα + c)2
Trigonometry (3A) 7 Young Won Lim1/2/15
Trigonometry in the 2nd 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
β = 180−α
Trigonometry (3A) 8 Young Won Lim1/2/15
Trigonometry in the 2nd 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
sinβ = (+ y ) / b
cosβ = (−x) / b
tanβ = (+ y )/(−x)
β = 180−α
Trigonometry (3A) 9 Young Won Lim1/2/15
Trigonometry in the 2nd Quadrant Angles (3)
(-x, y)
-x
y
a
c
b
A B
C
c
(-x, y)
-x-x
y
+r
r
r
sinβ = (+ y) / r
cosβ = (−x) / r
tanβ = (+ y) / (−x)
r = x2 y2
Isosceles Triangle
sin = y / b
cos = −x / b
tan = − y / x
Trigonometry (3A) 10 Young Won Lim1/2/15
Trigonometry in Quadrant Angles (4)
(-x, y)
-x
y
+r
sinβ = +sinα = (+ y)
cosβ = −sinβ = (−x )
tanβ = −tan α = (+ 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
Trigonometry (3A) 11 Young Won Lim1/2/15
Negative Angle Trigonometry (1)
y
x 1
(+x , + y )
−1
1st Quadrant Angle
sinα = (+ y)
cosα = (+x)
tanα = (+ y )/(+x)
−
−y
x
1
(+x , − y )
−1
4th Quadrant Angle
sin(−α) = − sinα = (−y )
cos (−α) = + cosα = (+x)
tan (−α) = − tanα = (−y )/(+x )
0 ̊ < α < 90 ̊ –90 ̊ < –α < 0 ̊
Trigonometry (3A) 12 Young Won Lim1/2/15
Negative Angle Trigonometry (2)
−
−y
−x
1
(−x , −y )
−1
3rd Quadrant Angle
sin(−β) = − sinα = (−y)
cos(−β) = − cosα = (−x)
tan (−β) = + tanα = (− y)/(−x)
sinβ = + sinα = (+ y )
cosβ = − cosα = (−x)
tanβ = − tanα = (+ y )/(−x)
y
−x 1
(−x , + y )
−1
2nd Quadrant Angle
90 ̊ < β < 180 ̊ –180 ̊ < –β < –90 ̊
Trigonometry (3A) 13 Young Won Lim1/2/15
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
sinθ = + sinα = (+ y)
cosθ = − cosα = (−x)
tanθ = − tanα = (+ y)/(−x)
= 180 ˚ −
90 ˚ 180 ˚
Trigonometry (3A) 14 Young Won Lim1/2/15
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
sinθ = − sinα = (−y)
cosθ = − cosα = (−x)
tanθ = + tanα = (−y)/(−x)
sinθ = − sinα = (−y)
cosθ = + cosα = (+ x)
tanθ = − tanα = (−y)/(+ x)
= − 180 ˚ = 360 ˚ −
−x x
270 ˚ 360 ˚180 ˚ 270 ˚
Trigonometry (3A) 15 Young Won Lim1/2/15
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
−x , y x , y
−x , − y x , − y
Trigonometry (3A) 16 Young Won Lim1/2/15
Trigonometric Functions
http://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 17 Young Won Lim1/2/15
Radian
http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 18 Young Won Lim1/2/15
Pythagorean identity
http://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 19 Young Won Lim1/2/15
Inverse Relations
http://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 20 Young Won Lim1/2/15
Symmetry
http://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 21 Young Won Lim1/2/15
Shifts and periodicity
http://en.wikipedia.org/wiki/Derivative
Trigonometry (3A) 22 Young Won Lim1/2/15
Making a Helix
Transparent OHP Film
Trigonometry (3A) 23 Young Won Lim1/2/15
y
z
y
z
x
x
z
y
x
Front View
Top View
Side View
A Helix and Viewpoints
Trigonometry (3A) 24 Young Won Lim1/2/15
y
z
y
z
y
x
Front View
Side View
0
2
32
Sine Wave
Sine Wave
Trigonometry (3A) 25 Young Won Lim1/2/15
y
z
y
x
Top View
Side View
0
2
32
y
z
Cosine Wave
Cosine Wave
Trigonometry (3A) 26 Young Won Lim1/2/15
Symmetry in Sinusoid
Trigonometry (3A) 27 Young Won Lim1/2/15
Sine and Cosine Waves
2− 312
32
52−
12 0
Sine Wave
Cosine Wave
Trigonometry (3A) 28 Young Won Lim1/2/15
Sine Wave Symmetry
2− 3
12
32
52−
12
0
Sine Wave
Trigonometry (3A) 29 Young Won Lim1/2/15
Cosine Wave Symmetry
2− 3
12
32
52−
12
0
Cosine Wave
Young Won Lim1/2/15
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