Trigonometry
36
θ sin
description
Trigonometry. θ. Definition of an angle. Terminal Ray. + counter clockwise. Initial Ray. - clockwise. Terminal Ray. Coterminal angles – angles with a common terminal ray. Terminal Ray. Initial Ray. Coterminal angles – angles with a common terminal ray. Terminal Ray. Initial Ray. - PowerPoint PPT Presentation
Transcript of Trigonometry
θsin
+ counter clockwise
- clockwise
Initial Ray
Terminal Ray
Terminal Ray
Definition of an angle
4
54
3
Coterminal angles – angles with a common terminal ray
Initial Ray
Terminal Ray
4
54
3
Coterminal angles – angles with a common terminal ray
Initial Ray
Terminal Ray
4
13
Radian Measure
2
,0
2
3
r
r1 Radian
57.3 o
2
360o = 2π radians
180o = π radians
Definition of Radians
C= 2πr
C= 2π radii
C= 2π radians
6
3
2
3
2
6
5
2,0
6
7
3
4
2
3
3
5
6
11
Unit Circle – Radian Measure
61
6
2
6
3
64
65
66
4
2
4
3
2,0
4
5
2
3
4
7
Unit Circle – Radian Measure
44
43
42
41
6
4
3
2
3
2
4
3
6
5
2,0
6
7
4
5
3
4
2
3
3
5 4
76
11
Unit Circle – Radian Measure
Degrees
Converting Degrees ↔ Radians
Recall oo
180,180
Converts degrees to Radians
o180Converts Radians to degrees
36
5
180
25
18025
oo
oo
50180
18
5
more examples
Trigonometric Ratios
θReference Angle
Adjacent Leg
HypotenuseOpposite Leg
hypotenuse
legoppositesin
hypotenuse
legadjacentcos
legadjacent
legoppositetan
Basic ratio definitions
2,0
Circle Trigonometry Definitions
(x, y)
Radius = r
Adjacent Leg = x
Opposite Leg = y
r
ysin
r
xcos
x
ytan
reciprocal functions
2,0
Unit - Circle Trigonometry Definitions
(x, y)
Radius = 1
Adjacent Leg = x
Opposite Leg = y
yy
1
sin
xx
1
cos
x
ytan
1
6
3
2
3
2
6
5
2,0
6
7
3
4
2
3
3
5
6
11
Unit Circle – Trig Ratios
2
1,
2
3
6
2
3,2
1
3
2
2
3,
2
1
3
2
2
1,
2
3
6
5
2,0
2
1,
2
3
6
7
2
3,
2
1
3
4
2
3
2
3,2
1
3
5
2
1,
2
3
6
11
Unit Circle – Trig Ratios sin cos tan
6
4
3
(+, +)
(-, -)
(-, +)
(+, -)
2
1
2
1
2
3
3
3
32
3
2
3
2
11
6
3
2
1
12
3
Reference AnglesSkip π/4’s
2
2,
2
2
4
2
2,0
2
3
2
2
2
2
1
Unit Circle – Trig Ratios sin cos tan
6
4
3
2
2
2
2 1
2
2,
2
2
4
3
2
2,
2
2
4
5
2
2,
2
2
4
7
(+, +)
(-, -)
(-, +)
(+, -)
2
2,0
2
3
Unit Circle – Trig Ratios sin cos tan
6
4
3
(+, +)
(-, -)
(-, +)
(+, -)
sin cos tan
2
2
3 -1
1
1
-1
0
0
0
0
0
0
Ø
Ø
(0, -1)
(0 , 1)
(1, 0)(-1, 0)
0 /2π
Quadrant Angles
View π/4’s
6
4
3
2
3
2
4
3
6
5
2,0
6
7
4
5
3
4
2
3
3
5 4
76
11
Unit Circle – Radian Measure
sin cos tan
6
4
3
2
1
2
1
2
3
3
3
32
3
2
2
2
2 1
(+, +)
(-, -)
(-, +)
(+, -)Degrees
1sin cos tan
2
2
3 -1
1
1
-1
0
0
0
0
0
0
Ø
Ø
0 /2π
Quadrant Angles
Graphing Trig Functions
f ( x ) = A sin bx
0 21
1
0
sin
sin)(
sin)(
f
xxf
0 21
1
0
cos
cos)(
cos)(
f
xxf
0 21
1
0
tan
tan)(
tan)(
f
xxf
Amplitude is the height of graph measured from middle of the wave.
Center of waveAmplitude
2
minmaxA
f ( x ) = A sin bx
0 21
1
0
cos
cos2
1)(
cos2
1)(
f
xxf
A = ½ , half as tall
f ( x ) = cos x
0 21
1
0
sin
sin2)(
sin2)(
f
xxfA = 2, twice as tall
f ( x ) = sin x
Period of graph is distance along horizontal axis for graph to repeat (length of one cycle)
bP
2
f ( x ) = A sin bx
0 21
1
0
sin
2
1sin)(
2
1sin)(
f
xxf
f ( x ) = sin x
B = ½ , period is 4π
0 21
1
0
cos
2cos)(
2cos)(
f
xxf
f ( x ) = cos x
B = 2, period is π
Special Right Triangles The PythagoreansGraphs Rene’ DesCartes Trigonometry Hipparchus, Menelaus, Ptolemy
2
2,0
2
3
Reference Angle Calculation
2nd Quadrant Angles
3rd Quadrant Angles4th Quadrant Angles
6
56
66
5
6
66
5
44
4
4
54
5
4
5
33
5
3
63
52
3
5
3
4
Return
Unit Circle – Degree Measure
0/360
150
240 300315
330
180
210
225
270
30
90
120
135 45
60
Return
Unit Circle – Degree Measure
0/360
150
240 300315
330
180
210
225
270
30
90
120
135 45
60
sin cos tan
2
1
2
1
2
3
3
3
32
3
2
2
2
2 1
(+, +)
(-, -)
(-, +)
(+, -)
30
45
60
Return
1sin cos tan
-1
1
1
-1
0
0
0
0
0
0
Ø
Ø
0/360
90
180
270
Quadrant Angles
o180
o36
180
36
5
o99 o180
180
99
20
11
10
3
o180
10
540o o54
12
5
o180 o75
Ex. # 6Ex. # 5
Ex. # 4Ex. # 3
return
2,0
Circle Trigonometry Definitions – Reciprocal Functions
(x, y)
Radius = r
Adjacent Leg = x
Opposite Leg = y
r
ysin
r
xcos
x
ytan
y
r
sin
1csc
x
r
cos
1sec
y
x
tan
1cot
return
6
4
3
2
3
2
4
3
6
5
2,0
6
7
4
5
3
4
2
3
3
5 4
76
11
2
2
2
2
2
3
2
1 1
1
Unit Circle – Radian Measure
1
