Mathematical Investigations III
Name
Mathematical Investigations III
Trigonometry- Modeling the Seas
SOLVING TRIGONOMETRIC EQUATIONS
For these problems, it will be helpful to recall your work on maximum and minimum points.
1.
2.
3.
Consider: sin(x) = 1.
a.
State one solution:
b.
How many solutions are there? How far apart are consecutive solutions?
c.
Combine this information to find the location of all solutions to this equation.
(Use k in your answer, and state the possibilities for it.)
Consider: sin(2x) = 1.
a.
State one solution:
b.
How many solutions are there? How far apart are consecutive solutions?
c.
Combine this information to find the location of all solutions to this equation.
Solve each of the following by finding a "starting" point and determining the distance
between consecutive values. (Note that "solve" requires finding all real solutions.)
a.
sin(x/3) = 1
b.
sin(2x) = 0
Trig. 11M.1
Rev. S05
Mathematical Investigations III
Name
3.
4.
5.
Continued.
c.
sin(3x) = –1
d.
sin(x/2) = 0
a.
Solve: cos(x) = 1. (Use the same process as above.)
b.
Solve: cos(3x) = 1.
Solve.
a.
cos(x/3) = 1
b.
cos(2x) = 0
cos(6x) = –1
d.
 2x 
cos    0
 3
b.
tan(x/4) = 0
c.
6.
Solve: tan(x) = 1
7.
Solve.
a.
tan(3x) = 1
Trig. 11M.2
Rev. S05
Mathematical Investigations III
Name
7.
8.
Continued.
tan(2x)  3
c.
d.
tan(x/5) + 1 = 0
1
.
2
Within the period [0, 2), how many solutions are there? State them:
Consider sin(x) 
a.
b.
State all solutions over the real numbers. (Note that a complete solution will
involve two "pieces.")
9.
Solve.
a.
10.
sin(x) 
3
2
b.
sin(x) 
 2
2
1
. We'll take a slightly different approach for this equation.
2
How does 2x relate to your solutions to problem 8?
Consider sin(2x) 
a.
b.
Now solve for x. (Check your answer for reasonableness. How far apart should
the answers be in each group?)
Trig. 11M.3
Rev. S05
Mathematical Investigations III
Name
11.
12.
Solve these problems with a phase shift.
  3

a.
sin  x   

6
2
b.
4  1

cos  x 


3  2
Solve.
a.
 x  1
sin   
 2 2
b.
cos 3x  
2
2
c.
sin 4 x  
3
2
d.
sin 2x  
 2
2
e.
sin 3x    
f.
3
 3x 
cos   
 2
2
g.
 x  5   3
cos 

 4 
2
h.
 x
2 cos    1  0
 3
1
2
Trig. 11M.4
Rev. S05