Transcript 5 3 Solving Trig Eqs I

• P. 369: 1-6 S
• P. 396: 7-20 (all solutions)
Objectives:
1. To solve trig
equations
Assignment:
• P. 369: 1-6 S
• P. 396: 7-20 (all
solutions)
When you solve an algebraic equation, your goal
is to isolate the variable:
2x  1
1
x
2
When you solve an trigonometric equation, your
goal is to isolate the trig expression:
2sin x  1
1
sin x 
2
Solve for x.
2sin x  1
1
sin x 
2
x  30 or

6
This is just one
possible solution, but
since sine is periodic,
that solution will come
back around again an
infinite number of
times. How do we get
the other solutions?
Solve for x.
2sin x  1
1
sin x 
2
x  30, 390, 750, etc.
x  30  360n
Solve for x.
2sin x  1
1
sin x 
2

13 25
x ,
,
, etc.
6
6
6
x

6
 2n
Solve for x.
2sin x  1
1
sin x 
2
But, wait, isn’t there another
angle that also has a sine of ½,
besides ones that are coterminal
with 30°?
x  150, 510, 870, etc.
x  150  360n
Solve for x.
2sin x  1
1
sin x 
2
But, wait, isn’t there another
angle that also has a sine of ½,
besides ones that are coterminal
with 30°?
5 17 29
x
,
,
, etc.
6
6
6
5
x
 2n
6
So the equation sin x = ½ has an infinite number
of solutions (since it’s periodic).
• If you find one solution, just add or subtract
the period to find some more
• If you were to put this into the calculator as
sin-1(1/2), your ignorant box of buttons will
only give you the solution between −π/2 and
π/2. How do you get the other set of
solutions?
For sine or cosecant, if one set of solutions start
with a, then the other set of solutions come
from 180° − a.
For cosine or secant, if one set of solutions start
with a, then the other set of solutions come
from 360° − a.
For tangent or cotangent, if one set of solutions
start with a, then the other set of solutions
come from 180° + a.
If a problem asks for all the possible solutions,
just find a solution, a, and then add the
period:
x  a  n  period 
For sine, cosine and
their reciprocals:
x  a  2n
For tangent and
cotangent:
x  a  n
Sometimes, however, a problem will ask for all
the solutions within a certain interval, like
(0, 2π].
In this case, just list out the answers in this
interval.
Solve for x.
2x 1  0
Solve for x.
2cos x 1  0
Solve for x.
3x  2  0
Solve for x.
3 csc x  2  0
Solve for x.
3x 2  4  0
Solve for x.
3sec2 x  4  0
Solve for x.
x  x  1  0
Solve for x.
sin x  sin x  1  0
Solve for x.



3x 2  1 x 2  3  0
Solve for x.



3 tan 2 x  1 tan 2 x  3  0
Solve for x.
2x2  x 1  0
Solve for x.
2 cos 2 x  cos x  1  0
Solve for x.
2 x 2  3x  1  0
Solve for x.
2sin 2 x  3sin x  1  0
Objectives:
1. To solve trig
equations
Assignment:
• P. 369: 1-6 S
• P. 396: 7-20 (all
solutions)