10 - 4 Solving trig equations
Try the quiz at the bottom of the page!
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When solving trig equations, there are some rules you should try to follow:
1)  Remember your basic factoring!!
2)  Do not divide by a function if it divides out on both sides!
3)  If the equation involves functions of 2x and x, use the identities to write in terms of functions of x.
4)  If the equation involves functions of 2x only, solve directly for 2x and then solve for x.
5)  You can use a graphing calculator to find where they intersect.
Remember to watch for the domain in the problem and make sure your answers fit in that domain!

sample problems
1)  Solve for   cos 2x + sin x = 1  for 0o < x < 360o
Solution:
Write cos 2x in the form with sine only!!  Why?  (Other function involves sine)
1 - 2sin2x + sin x = 1
Set one side equal to 0
0 = 2sin2x - sin x
Factor!
0 = sin x(2sin x - 1)
sin x = 0 or 2sin x - 1 = 0
sin x = 0 or sin x = 1/2
x = sin-10 or x = sin-1(1/2)
x = 0o, 180o, 30o, 150o

2)  Solve for  tan 2x = tan x  for 0o < x < 360o
Solution:
Use the identity for double angle for tangent!

3)  Solve for sin 2x = cos 2x  for 0o< x < 360o
Solution:

Since both functions are written in 2x, solve directly!
divide both sides by cos 2x.  This will not cancel on the left side!

4)  Solve for tan(x - 20o) = 1  for 0o < x < 360o
Solution:
Solve directly!!
x - 20o = tan-11
x - 20o = 45o, 225o
x = 65o, 245o

5)  Solve  3cos 2x + sin x = -2  for 0o < x < 360o
Solution:

Use the double angle formula for cosine involving just the sine!
3(1 - 2sin2x) + sin x = -2
3 - 6sin2x + sin x = -2
Set one side equal to zero!
0 = 6sin2x - sin x - 5
Factor!
0 = (6sin x + 5)(sin x - 1)
sin x = -5/6 or sin x = 1
sin x = 1 means x = 90o
The reference angle for the first solution is x = 56.4o Since we need the number negative, the answers are in the III and IV quadrants.
x = 180 + 56.4 = 236.4o or x = 360 - 56.4 = 303.6o
x = 90o, 236.4o, 303.6o

6)  Solve using the calculator for 0 < x < 2p      3sin 2x = 1
Solution:
Graph both y = 3 sin 2x and y = 1 through the first cycle!


 
You can see from the graph that they intersect at 4 places between 0 and 2p.
Using your calculate button on the TI-82, find the four intersection points.  We are interested in the x values, not the y values.
x = .17, 1.40, 3.31, 4.54
Hopefully, you are ready for the sample test.  If not, hit the previous button.  If so, go for the next button!



Current quizaroo #  10

1)  Simplify:  cos(p - x)


a)  cos x
b)  -cos x
 c)  1
 d)  -1
 e)  0
 
 
 


2)  Find:   tan (a + b) if  tan a = 3/2 and tan b = 1/2

          a)  2

b)  4
c)  8
d)  1
e)  0
 
3)  If the sin x = 4/5, find the sin 4x.  x is an acute angle.
 
a)  24/25
    b)  336/625
c)  16/25
                                                                                d)  3/5
                                                                                 e)  1
 
 
 
4)  Simplify   (1 - cos 2x)/(1 + cos 2x)
 
a)  sin22x
b)  cos22x
c)  cot22x
d)  tan22x
e)  sec22x

  5)  Solve:  cos 2x = cos x   for  0 < x < 2p

a)  p/6, 5p/6
b)  p/2, 3p/2, p/6, 5p/6
c)  0, 1
d)  0, 2p/3, 4p/3
e)  p, 2p/3, 5p/6
 
 

 click here for answers!!