Transcript Document

Trigonometric
Equations (II)
In this section we'll learn various techniques to manipulate
trigonometric equations so we can solve them. We'll find
solutions on the interval from 0 to 2. The first tip is to try
using identities to get in terms of the same trig function.
cos   sin   sin   0
2
2
Use the Pythagorean Identity to replace this with an
2
equivalent expression using sine.
cos   1  sin 
1  sin   sin   sin   0
2
2
2sin   sin   1  0
2
(2sin   1)(sin   1)  0
1
sin    , sin   1
2
2
Combine like terms,
multiply by -1 and put
in descending order
Factor (think of sin  like x and
this is quadratic)
Set each factor = 0 and solve
7 11 

,
,
6
6 2
When we don't have squared trig functions, we can't use the
Pythagorean identities. If you have two terms with different
trig functions you can try squaring both sides.
cos   sin   0 2
2
Square both sides. Must do whole
side together NOT each term (so
left side will need to be FOILed).
cos   2cos sin   sin   0
2
2
re-order terms
cos sin
sin   cos 122cos
sin 00
2
2
Pythagorean Identity---this equals 1
1  sin 2  0
sin 2  1
3
3
2 
so  
2
4
Double angle Identity
sin 2
Get sine term
alone
Where is the sine -1?
Remember to do
another loop when
you have 2
 2sin  cos 
7
7
2 
so  
2
4
HELPFUL HINTS FOR SOLVING
TRIGONOMETRIC EQUATIONS
•Try to get equations in terms of one trig function by using
identities.
•Be on the look-out for ways to substitute using identities
•Try to get trig functions of the same angle. If one term is
cos2 and another is cos for example, use the double angle
formula to express first term in terms of just  instead of 2
•Get one side equals zero and factor out any common trig
functions
•See if equation is quadratic in form and will factor. (replace
the trig function with x to see how it factors if that helps)
•If the angle you are solving for is a multiple of , don't forget
to add 2 to your answer for each multiple of  since  will
still be less than 2 when solved for.
There are some equations that can't be solved by hand
and we must use a some kind of technology.
Use a graphing utility to solve the equation. Express any solutions
rounded to two decimal places.
22 x  17 sin x  3
Graph this side as y1
in your calculator
Graph this side as y2
in your calculator
You want to know where they are equal. That would be
where their graphs intersect. You can use the trace
feature or the intersect feature to find this (or these)
points (there could be more than one point of
intersection).
22 x  17 sin x  3
This
This was
is offgraphed
a little due
on
the
to the
computer
fact we with
check: 22 .53 17sin .53  3.066  3
graphcalc,
approximated.
a freeIf you
graphing
carried it utility
to more
you
can
decimal
download
placesatyou'd
www.graphcalc.com
have more accuracy.
After seeing the initial
graph, lets change the
window to get a better
view of the intersection
point and then we'll do
a trace.
Rounded to 2 decimal places, the
point of intersection is x = 0.53
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au