Download 4.4 Introduction to Trigonometric Equations

Write the coordinates P(x, y)
Factor
x2  5x  6
 x  3 x  2 
3 x 2  14 x  5
 3x  1 x  5
 
sin  
2
1
sin  
2
 
2  
sin    cos  
6
6
2
sin 2   5sin   6  0
Trigonometric Expressions
Trigonometric Equations
Rewrite the equation to isolate the trig function. Solve for the
angle within the given domain. Write the general solution.
a ) 2sin   1  0
2sin   1
1
sin  
2
0    360
b) 4 cos   1  cos 
3cos   1
y is positive
I or II
special ratio
ref angle 30°
  30 or 150
1
cos   
3
0    2
x is negative
II or III
not special ratio
ref angle 1.23
  1.91 or 4.37
  30  360 n, n  I
  1.91  2 n, n  I
  150  360 n, n  I
  4.37  2 n, n  I
c)
3 csc   2  0 0    360
3 csc   2
2
csc   
3
3 y is negative
sin   
2
III or IV
0    2
d ) 4sec   3  3sec   2
sec   1
cos   1
x is - 1
quadrantal angle
special ratio
ref angle 60°
  2400 or 300
  240  360 n, n  I
  300  360 n, n  I
 
    2 n, n  I
Solve for  if 0 ≤  < 2.
1
2
a) sin  
4
b) cs c  
3
2
2
1
sin   
2
1
sin   
2
y is negative
or positive
2
sin   
2
Reference
r
Angle


4
csc   
3
I, II,
III or IV

4
 3 5 7
4
,
4
,
4
,
4
2
csc   
3
y is negative
or positive
I, II,
3
III or IV
sin   
2

Reference
r 
Angle
3
 2 4 5
 ,
,
,
3 3
3
3
Solving Trigonometric Equations
c) 4sin2 x - 3 = 0
4sin2 x = 3
 2 4 5
x
,
,
,
3 3 3
3
3
sin x 
4
2
3
sin x  
4
3
sin x  
2
Reference
Angle
xr 
0 ≤  < 2.

3

y is negative
or positive
I, II,
III or IV

4


2
n, n  I
0 ≤  < 2.
Solving Second Degree Trigonometric Equations
d) 2 sin2   sin   0
sin  (2sin   1)  0
sin   0 or 2sin   1  0
quadrantal
1
sin   
2

Reference


r
Angle
6
  0, , 7  , 11
6
6
y is negative
III or IV
e) 2 sin2   sin  1  0
(2 sin  1)(sin  1)  0
2 sin  1  0 or sin  1  0
1
sin  
or sin   1
2

Reference


r
Angle
6

 5 3
,
,
6 6
2
Solving Trigonometric Equations
0 ≤  < 2.
f) 2 cos2   7cos   3  0
(2 cos   1)(cos   3)  0
2 cos   1  0 or cos   3  0
1
cos   3
cos  
or
Reference
Angle
2


3

 5
,
3 3
NO solution for
cos  = 3.
0 ≤ x < 360°
g)
(sin x  1)(tan x  1)  0
sin x  1  0
tan x  1  0
sin x  1
tan x  1
x  90
x  45
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1b, 2, 3, 4a,b, 5, 7, 9, 10, 11, 12, 18, 19