sullivan algebra and trigonometry: section 6.4 trig functions of general angles objectives of this...
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Sullivan Algebra and Trigonometry: Section 6.4
Trig Functions of General AnglesObjectives of this Section
• Find the Exact Value of the Trigonometric Functions for General Angles
• Determine the Sign of the Trigonometric Functions of an Angle in a Given Quadrant
• Use Coterminal Angles to Find the Exact Value of a Trigonometric Function
• Find the Reference Angle of a General Angle
• Use the Theorem of Reference Angles
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Let be any angle in standard position, and
let denote the coordinates of any point,
except the origin (0, 0), on the terminal side
of . If denotes the distance from
(0, 0) to ( , then the
are defined as the ratios
a b
r a b
a b
,
, )
2 2
six trigonometric
functions of
sin cos tan
csc sec cot
b r a r b a
r b r a a b
provided no denominator equals 0.
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(a, b)
rx
y
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Find the exact value of each of the six trigonometric functions of a positive angle if (-2, 3) is a point on the terminal side.
(-2, 3)
x
y
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a b 2 3,
r a b 2 2 2 22 3 13( )
sin br
313
3 1313
cos ar
213
2 1313
tan
ba
32
32
csc rb
133
sec ra
132
cot ab
23
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P= (1, 0)
sin sin0 001
0 br
cos cos0 011
1 ar
tan tan0 001
0 ba
csc csc0 010
rb
sec sec0 011
1 ra
cot cot0 010
ab
P= (a, b)
x
y
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P= (0,1)
sin sin2
9011
1 br
cos cos2
9001
0 ar
tan tan2
9010
ba
csc csc2
9011
1 rb
sec sec2
9010
ra
cot cot2
9001
0 ab
x
y
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sin
cos
tan
csc
sec
cot
0
1
0
1
Not defined
Not defined
180( radians)
1
0
1
Not defined
Not defined
0
270 3 2( radians)
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x
y
(a, b)
a < 0, b < 0, r > 0
r
a > 0, b > 0, r > 0a < 0, b > 0, r > 0
a > 0, b < 0, r > 0
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I (+, +)
All positive
II ,
sin , csc 0 0
All others negative
III ,
tan , cot 0 0
All others negative
IV ,
cos , sec 0 0
All others negative
x
y
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Two angles in standard position are said to be coterminal if they have the same terminal side.
x
y
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Let denote a nonacute angle that lies in a quadrant. The acute angle formed by the terminal side of and either the positive x-axis or the negative x-axis is called the reference angle for .
Reference Angle
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Finding the reference angle
1.
.
Add / subtract multiples of 360 2
until you obtain an angle between
0 and 360 0 and 2 radians
2. Determine the quadrant in which the terminal side of the angle formed by the angle lies.
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180
180
360
2
x
y
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Reference Angles
If is an angle that lies in a quadrant and if
is its reference angle, then
sin sin csc csc tan tan cos cos
cot cot sec sec
where the + or sign depends on the
quadrant in which lies.
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Find the exact value of each of the following trigonometric functions using reference angles:
(a) cos 570 (b) tan16
3
(a) 570 360 210 in Quadrant III, so cos < 0
210 180 30
cos cos210 303
2
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b 16
3 2
163
63
103
103
63
43
is in Quadrant III, so tan > 0
43 3
tan tan16
3 33
2