This document discusses solving trigonometric equations graphically. It provides examples of using the intersection method and x-intercept method to find approximate solutions to trig equations by graphing them. Key steps outlined are writing the equation in the form f(x)=0, determining the period, graphing over the interval, and finding solutions graphically. It notes that trig equation solutions can also be expressed in degrees. Examples are provided for solving equations like sin(x)=a and tan(x)=b graphically in both radians and degrees.

1. 8.1: Graphical Solutions to
Trigonometric Equations
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Solve trig equations
graphically.
•State the complete
solution of a trig equation

2. Example
Find the approximate
solutions to the following
equation using the
intersection method:
2 3
3 5 2x x x x− − = − −

3. Example
Find the approximate
solutions to the following
equation using the x
intercept method:
2 3
4 3 6x x x x− − = + −

4. Important Idea
Many of the methods and
ideas we learned about
algebraic equations also
work with trigonometric
equations.

5. Example
1
cos
2
x =
Solve by
graphing:
1 cosy x=2. Graph
3. Graph 2
1
2
y =
4. Find window
5. Use 2nd
Calc 5 to
find 2 solutions
6. Use Period to
find all solutions
1. Set mode to rads
using
intersect
ion
method

6. Important Ideaand
always have 2
positive or 2
negative values
between 0 and
sin x cosx
2π . Find the 1st
2 solutions after
0 with your
calculator.
2π

7. Try This
tan 2x =
1.1071x kπ= +
Solve by
graphing: 2

8. Important Idea
will have
only 1 solution
between 0 and
tan x
(its period).
Find the 1st
solution after 0;
all other
solutions are
repeats.
π
0 π

9. Example
Solve
sin .85x = −
1. Set mode to rads
2. Re-write
3. Graph
1 sin .85y x= +
4. Find window
5. Find 2 solutions
using 2nd
calc 2
6. Use period to
find all solutions
using the
x-
intercept
method

10. Try This
Solve using
the x
intercept
method:
cos .75x = − 2.419 2x kπ= +
3.864 2x kπ= +
and

11. Example
Solve
2
3sin cos 2 0x x− − =
using the x
intercept method.
Hint: find all
solutions within
the period and add
2 kπ to each.

12. Important Idea
1. Write in form ( ) 0f x =
2. Determine period, p
3. Graph over interval p
4. Graphically, find all
solutions in interval p
5. Add to solutionkp
To solve any trig equation:

13. Example
Solve
tan 3sin2x x=

14. Try This
Solve
tan 4cos2x x= −
0.6830
0.9763
1.2801
x k
x k
x k
π
π
π
= − +
= +
= +
3 non-
repeating
solutions

15. Important Idea
Some real-world
applications require that
we solve trig equations in
degrees instead of
radians.

16. Try This
What two things are done
differently if we solve in
degrees instead of radians?

17. Example
Solve in degrees:
2
2sin 3sin 3 0θ θ− − =

18. Lesson Close
In a future lesson, we will
learn to solve
trigonometric equations
algebraically.