Solving Equations
Lesson 1

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Pre – Calculus Math 40S: Explained!
www.math40s.com
136
Trigonometry 2 - Lesson 1
Part One - Graphically Solving Equations
Solving trigonometric equations graphically:
When a question asks you to solve a system of trigonometric equations, they are looking for the
values of
θ
that make both equations true. There are two ways you can solve for
θ
: graphically
in your TI-83, and algebraically. Part I will show the graphing method, and Parts II & III will
focus on algebraic methods.
Example 1:
Solve
1
cos =
2
θ
and state the general solutions:
In your TI-83, graph each
equation in degree mode.
Example 2:
Solve
2
cos2 =
2
θ
and state the general solutions:
Graph both equations in your TI-83,
then solve for the first two intersection points.
The first two intersection points are at 22.5º
and 157.5º. As you can see in the graph, the
solutions repeat themselves every period.
Since the
b
-value is 2, the period is 180º, or
π
.
The first general solution is:
22.5º ±
n
(180º) or
8
n
π
π
±
And the second is:
157.5º ±
n
(180º) or
7
8
n
π
π
±
Now use 2
nd
Æ
Trace
Æ
Intersect to find the
points of intersection. They occur at 60º & 300º
If you extend the window, you will see that the
intersection points are in the same relative places,
one period later.
The first general solution is:
60º ± n(360º)
or
(2
)
3
n
π
π
±
and the second is:
300º ± n(360º)
or
5
(2
)
3
n
π
π
±

Pre – Calculus Math 40S: Explained!
www.math40s.com
137
Trigonometry 2 - Lesson 1
Part One - Graphically Solving Equations
Example 3:
Graphically find the general solutions for
2sin - 3
θ
= 0
Note that even if you manipulate the equation, you can still solve by graphing:
If you re-arrange the equation to
2sin
=
3
θ
by taking
3
to the other side, we get:
If we manipulate the equation again by dividing both sides by 2, we get:
3
sin =
2
θ
.
Solving this:
Solving, we still have the same answers of 60º & 120º
Graph the two equations in your TI-83 and
solve by finding the points of intersection.
60º ± n(360º)
or
(2
)
3
n
π
π
±
120º ± n(360º)
or
2
(2
)
3
n
π
π
±
(In this case, another method would be to find
the x-intercepts using 2
nd
Æ
Trace
Æ
Zero)
Once again, we still get the same answers of 60º & 120º
Manipulating an equation does NOT
change the solution!

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