Download 6.1.1 Solving Trig Equations

6.1.1 SOLVING TRIG
EQUATIONS
FRIDAY, FEBRUARY 13, 2015
INTRODUCTION
• You can already solve all kinds of equations. You
have been doing it since your first algebra class.
• You know that some equations have no solutions,
some have only one solution, and others can have
any number of solutions, even an infinite number.
x=x+1
x+1=x+1
5=x+1
INTRODUCTION
• Today’s lesson focuses on solving trigonometric
equations (which we will refer to as trig equations
from now on).
• The concepts and the procedures you will use are
basically the same, but there are a few unique
qualities of trig equations that you have to keep in
mind.
• Let’s see what sets this kind of solving apart from the
rest.
• 6-1. We begin with the graphs of y = sin x and y = 12
shown below. For now, concentrate on the portion
of the graph shown, 0 ≤ x ≤ 2π .
• How do you know that the equation, sin x = 12 has
more than one solution?
• What are the solutions for sin x = 12 , where 0 ≤ x ≤ 2π ?
6.1 CONTINUED
• Now recall the unit circle. The horizontal line y = 12
has been drawn across the circle. Explain how this
can help to find solutions to the equation sin x = 12
for 0 ≤ x ≤ 2π .
6.1 CONTINUED
• Using both the graph and the unit circle, we want
to solve the equation sin x = − 23 .
• How?
• 6-2. What about cos x = 12 over the same interval,
0 ≤ x ≤ 2π ? Draw a line on the unit circle so that
you can find where the x-coordinate equals. What
angles satisfy the equation?
CLOSURE
• Solving a trigonometric equation is like solving any
equation: our goal is to get the variable by itself on
one side of the equal sign.
• We want to find out what values of the variable will
make the equation “true.”
• But, we need to keep in mind when solving trig
equations, that there are not necessarily a finite
number of solutions.
• The solutions we want will depend on the problem
situation.
• If only a few solutions are required, we can limit the
domain to a particular interval.
MATH NOTES - SOLVING PERIODIC
FUNCTIONS
• When solving periodic functions (such as sin x = − 22 ),
we get an infinite number of solutions (or no
solution).
• In the example sin x = − 22 , we get the solutions:
5𝜋 13𝜋
𝜋 7𝜋 15𝜋
x = … −3𝜋
,
,
…
and
x
=
…
−
,
, 4 …
4
4
4
4 4
• This may be written more compactly as x = 5𝜋
+ 2πn,
4
7𝜋
+ 2πn, where n is any integer.
4
• If we restrict the domain to [0, 2π ), we get only two
7𝜋
solutions: x = 5𝜋
and
.
4
4