5.3
Solving Trig. Equations
2 sin x - 1 = 0
2 sin x = 1
sin x = 1/2
X=
!
+ 2k!
6
5!
X=
+ 2k!
6
1
2
2
1
sin x +
2
= - sin x
2 sin x = ! 2
sin x = !
2
2
-1
-1
2
5!
x=
+ 2k!
4
x=
7!
+ 2k!
4
2
3 tan2 x - 1 = 0
tan2 x = 1/3
1
tan x = ±
3
1
-1
!
+ k!
x=
6
5!
x=
+ k!
6
! 3
3
1
-1
cot x cos2 x = 2 cot x
cot x cos2 x - 2 cot x = 0
cot x(cos2 x - 2) = 0
cot x = 0
!
x=
+ k!
2
1
0
-1
cos2 x - 2 = 0
cos x =
± 2
undefined
Find all solutions of 2 sin2 x - sin x - 1 = 0 in
2 sin2 x - sin x - 1 = 0
(2sin x + 1)( sin x - 1) = 0
sin x = - 1/2
and
sin x = 1
7!
11!
!
x=
,x =
,x =
6
6
2
[0,2! )
Writing in terms of a single trigonometric function.
2 sin2 x + 3 cos x - 3 = 0
2(1 - cos2 x) + 3 cos x - 3 = 0
2 - 2 cos2 x + 3 cos x -3 = 0
1
1
0 = 2 cos2 x - 3 cos x + 1
0 = (2 cos x - 1)(cos x - 1)
2
1
cos x = , cos x = 1
2
General solutions:
1
2
!
5!
x = + 2k! , x =
+ 2k! , x = 2k!
3
3
Find all solutions of 2 cos 3t - 1 = 0
2 cos 3t - 1 = 0
cos 3t = 1/2
Where does cos t = 1/2 at?
!
5!
+ 2k! ,
+ 2k!
3
3
!
3t = + 2k!
3
! 2k!
t=
9
+
3
Now, set each of these = to 3t
5!
3t =
+ 2k!
3
5! 2k!
t=
+
9
3