Trigonometry

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Trigonometry
?
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Definition of an angle
Terminal Ray
Counter clockwise
Initial Ray
- clockwise
Terminal Ray
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Coterminal angles angles with a common terminal
ray
Terminal Ray
Initial Ray
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Coterminal angles angles with a common terminal
ray
Terminal Ray
Initial Ray
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Radian Measure
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Definition of Radians
C 2pr
C 2p radii
C 2p radians
360o 2p radians
r
180o p radians
1 Radian ? 57.3 o
r
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Unit Circle Radian Measure
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Unit Circle Radian Measure
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Unit Circle Radian Measure
Degrees
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Converting Degrees ? Radians
Converts degrees to Radians
Recall
Converts Radians to degrees
more examples
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Trigonometric Ratios
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Basic ratio definitions
Hypotenuse
Opposite Leg
Reference Angle
?
Adjacent Leg
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Circle Trigonometry Definitions
(x, y)
Radius r
Opposite Leg y
Adjacent Leg x
reciprocal functions
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Unit - Circle Trigonometry Definitions
(x, y)
Radius 1
Opposite Leg y
Adjacent Leg x
1
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Unit Circle Trig Ratios
sin
cos
tan
(, )
(-, )
(, -)
(-, -)
Skip p/4s
Reference Angles
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Unit Circle Trig Ratios
sin
cos
tan
(, )
(-, )
(-, -)
(, -)
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Unit Circle Trig Ratios
sin
cos
tan
(, )
(-, )
(0 , 1)
Quadrant Angles
(1, 0)
(-1, 0)
sin
cos
tan
0
/2p
1
0
0
0
Ø
1
(0, -1)
(-, -)
(, -)
0
-1
0
Ø
-1
0
View p/4s
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Unit Circle Radian Measure
sin
cos
tan
(-, )
(, )
Quadrant Angles
sin
cos
tan
1
0
/2p
1
0
0
0
Ø
1
(-, -)
(, -)
0
-1
0
Ø
Degrees
-1
0
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Graphing Trig Functions
f ( x ) A sin bx
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Amplitude is the height of graph measured from
middle of the wave.
Amplitude
Center of wave
f ( x ) A sin bx
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f ( x ) cos x
A ½ , half as tall
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f ( x ) sin x
A 2, twice as tall
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Period of graph is distance along horizontal axis
for graph to repeat (length of one cycle)
f ( x ) A sin bx
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f ( x ) sin x
B ½ , period is 4p
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f ( x ) cos x
B 2, period is p
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A unit circle is a circle with a radius of 1
unit. For every point P(x, y) on the unit circle,
the value of r is 1. Therefore, for an angle ? in
the standard position
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The End
Trigonometry Hipparchus, Menelaus, Ptolemy
Special Right Triangles The Pythagoreans
Graphs Rene DesCartes
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Reference Angle Calculation
2nd Quadrant Angles
3rd Quadrant Angles
4th Quadrant Angles
Return
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Unit Circle Degree Measure
90
120
60
135
45
150
30
0/360
180
330
210
225
315
300
240
270
Return
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Unit Circle Degree Measure
sin
cos
tan
30
90
(-, )
(, )
45
120
60
135
45
60
150
30
Quadrant Angles
0/360
180
sin
cos
tan
1
330
210
0/360
1
0
0
225
315
0
Ø
1
90
300
240
(-, -)
(, -)
0
-1
0
180
270
Ø
Return
-1
0
270
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Ex. 4
Ex. 3
Ex. 6
Ex. 5
return
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Circle Trigonometry Definitions Reciprocal
Functions
(x, y)
Radius r
Opposite Leg y
Adjacent Leg x
return
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Unit Circle Radian Measure
1