Euclidean geometry and trigonometry 1 Euclidean geometry means flat space sine and cosine Calculating Trigonometric identities A C M E 2 1 q x y
Feb 09, 2016
Euclidean geometry and trigonometry
1
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2π
1 πΌ
π½
q x
y
Euclidean geometry
2
BA
D
C
(1) Line segment
(2) Extend line segment into line
F
E
(3) Use line segment to define circle
(4) All right angles are equal
(5) Parallel postulate
Euclidean geometry: Flat space
3
(5) Parallel postulate
Flat
Curved
Non-embeddable spaces(Cannot be drawn as rippled surfaces in higher-dimensional flat spaces)
Euclidean geometry: Pythagorean theorem
4
a
bc
Euclidean geometry: Pythagorean theorem
5
a
bcc2
a2
b2
a2 + b2 = c2Want to show
Euclidean geometry: Pythagorean theorem
6
a
bc
b
a2 + b2 = c2Want to show
(a β b)2
ab/2
ab/2
ab/2
ab/2
(a - b)2 + 4ab/2 = c2
a2 -2ab + b2 + 2ab = c2
a2 + b2 = c2
π=βπ2+π2
a - b
Euclidean geometry and trigonometry
7
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2π
1 πΌ
π½
q x
y
Trigonometry: sine and cosine
8
1
qx
y
q
Trigonometry: sine and cosine
9
1
x = cos(q)
qx
y
y = sin(q)
2π
ACME
10
2ππ
0
-1
1
ππ4
π2
3π2
12
β1 /2
β2/2β3 /2
ββ3 /2ββ2 /2
3π4
5π4
7π4
π6
π3
2π35π6
7π6
4π3
5π311π6
sin (π )
cos (π )
x
y
Trigonometry: sine and cosine
-1
1
12
β1 /2
β2/2β3 /2
ββ3 /2ββ2 /2
sin (π )
cos (π )
11
Trigonometry: sine and cosine
2π0
ππ4
π2
3π2
3π4
5π4
7π4
π6
π3
2π35π6
7π6
4π3
5π311π6
π
Euclidean geometry and trigonometry
12
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2π
1 πΌ
π½
q x
y
Trigonometry:
13
Want to approximate
π3
π3π
3
π3
π3
1 1
1
1 1
π3
π3
π3
π3
11
ACME
2π
1
Trigonometry:
14
Want to approximate
π3
π3
π3
1
1
1
π3
π3
π3
1
1
1
Trigonometry:
15
Want to approximate
π6π6
1/2 1/2
x
( 12 )2
+π₯2=12
π₯2=1β 14
π₯=β 34=β32
β32
π3
π3
1 1
Trigonometry:
16
Want to approximate
π6π6
1/2 1/2
x
( 12 )2
+π₯2=12
π₯2=1β 14
π₯=β 34=β32
β32
β32
1/2
1
π6
Trigonometry:
17
Want to approximate
( 12 )2
+π₯2=12
π₯2=1β 14
π₯=β 34=β32
β32
1/2
1
π6
Trigonometry:
18
Want to approximate
β32
1/2
1
π6
π6
1β β32
1
y
( 12 )2
+(1β β32 )
2
=π¦2
14+( 2ββ3
2 )2
=π¦2
2ββ3=π¦2
π¦=β2ββ3π6 β³ β2ββ3
πβ³6 β2ββ3πβ³3.1058
STOP
Trigonometry:
19
πβ³3.1058
Sine! Sine!Cosine,
Sine!
3 . 1415 9!
ACME
2π
1
-1
1
12
β1 /2
β2/2β3 /2
ββ3 /2ββ2 /2
sin (π )
cos (π )
20
Trigonometry: sine and cosine
1 Β°β π180
0.5240.785
1.0471.571
2.0942.356
2.6183.142
3.6653.927
4.1895.236
4.7125.498
5.7606.283
30Β°45Β°
60Β°90Β°
120Β°135Β°
150Β°180Β°
210Β°225Β°
240Β°300Β°
270Β°315Β°
330Β°360Β°
2π0
ππ4
π2
3π2
3π4
5π4
7π4
π6
π3
2π35π6
7π6
4π3
5π311π6
π
Euclidean geometry and trigonometry
21
Euclidean geometry means flat space sine and cosine
Calculating Trigonometric identities
ACME
2π
1 πΌ
π½
q x
y
Trigonometry: sine and cosine in terms of right triangles
22
1
x = cos(q)
qx
y
y = sin(q)
q
q
Trigonometry: sine and cosine in terms of right triangles
23
1
q
sin(q
)
cos(q)
r
r cos(q)
r sin
(q)
R
R sin
(q)
R cos(q)
Proving identities: Pythagorean identity
24
STOP cos2 (π )+sin2 (π )=?
cos2 (π )+sin2 (π )=1Pythagorean identity
1
q
sin(q
)
cos(q)
Proving identities: Angle addition formula
25
πΌ
Want to show
sin (πΌ+π½ )=sin (πΌ ) cos (π½ )+cos (πΌ ) sin (π½ )π½
1
sin ( π½)
cos(π½ )
π2 β π½π
2 βπΌ πΌ+π½
sin(πΌ
+π½
)
sin (π½ )π₯ =
h sin (πΌ )hcos (πΌ )
h
xsin( π½)cos
(πΌ )
sin(πΌ )
Proving identities: Angle addition formula
26
πΌ
Want to show
sin (πΌ+π½ )=sin (πΌ ) cos (π½ )+cos (πΌ ) sin (π½ )π½
sin(πΌ
+π½
)
sin( π½)cos
(πΌ )
sin(πΌ )
(sin (π½) cos (πΌ )sin (πΌ )
+cos ( π½))sin (πΌ )=ΒΏ sin (πΌ+π½ )ΒΏ
cos(π½ )