7.4 Trigonometric Functions of General Angles What if the angle is not acute?
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Transcript 7.4 Trigonometric Functions of General Angles What if the angle is not acute?
7.4
Trigonometric Functions
of General Angles
What if the angle is not acute?
Let be any angle in standard position, and
let a , b denote the coordinates of any point,
except the origin (0, 0), on the terminal side
of . If r a b denotes the distance from
(0, 0) to (a , b), then the six trigonometric
functionsof are defined as the ratios
sin b r
cos a r
tan b a
csc r b
sec r a
cot a b
2
2
provided no denominator equals 0.
y
r
(a, b)
x
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.
y
(-2, 3)
x
a 2, b 3
r a 2 b2 ( 2) 2 32 13
b
3 3 13
sin
r
13 13
r
13
csc
b
3
a 2 2 13
r
13
cos
sec
a
2
r
13
13
b 3
3
tan
a 2
2
a
2
cot
b
3
y
r 1
P= (1, 0)
P= (a, b)
x
b 0
sin 0 sin 0 0
r 1
a 1
cos 0 cos 0 1
r 1
b 0
tan 0 tan 0 0
a 1
r 1
csc 0 csc 0
b 0
r 1
sec 0 sec 0 1
a 1
a 1
cot 0 cot 0
b 0
sin
y
cos
2
tan
P= (0,1)
x
2
csc
sec
cot
2
2
2
2
sin 90
cos 90
tan 90
csc 90
sec 90
cot 90
b 1
1
r 1
a 0
0
r 1
b 1
a 0
r 1
1
b 1
r 1
a 0
a 0
0
b 1
180 ( radians)
270 (3 2 radians)
sin
0
1
cos
tan
1
0
0
Not defined
csc
Not defined
1
sec
1
Not defined
cot
Not defined
0
y
a < 0, b > 0, r > 0
a > 0, b > 0, r > 0
r
x
(a, b)
a < 0, b < 0, r > 0
a > 0, b < 0, r > 0
y
II ,
sin 0, csc 0
All others negative
III ,
tan 0, cot 0
All others negative
I (+, +)
All positive
IV ,
x
cos 0, sec 0
All others negative
y
II ,
I (+, +)
Students
All
( Sin )
III ,
(All functions)
IV ,
Take
Care
( Tangent )
( Cosine )
x
Two angles in standard position
are said to be co-terminal if they
have the same terminal side.
y
x
Let denote a non-acute 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
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.
y
180
or
x
180
or
360
or
2
Theorem: Reference Angles
If is an angle that lies in a quadrant and if
is its reference angle, then
cos cos
sin sin
csc csc
tan tan
cot cot
sec sec
where the + or sign depends on the
quadrant in which lies.
Find the exact value of each of the
following trigonometric functions using
reference angles:
16
(a) cos 570
(b) tan
3
(a) 570 360 210
in Quadrant III, so cos < 0
210 180 30
3
cos 210 cos 30
2
16
16 6 10
2
b
3
3
3
3
10 6 4
3
3
3
is in Quadrant III, so tan > 0
4
3
3
16
3
tan
tan
3
3 2