Solving Trigonometric Equations
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Transcript Solving Trigonometric Equations
Solving
Trigonometric Equations
Solving
Trigonometric Equations
For most problems,
The solution interval
Will be
[0, 2)
You are responsible for checking your solutions back into the original problem!
First Degree Trigonometric
Equations:
• These are equations where there is one
kind of trig function in the equation and
that function is raised to the first power.
2 sin(x) 1
2 sin(x) 1
1
sin( x)
2
Now figure out where sin = -1/2 on the unit circle.
1
7
11
sin
at
and
2
6
6
Complete the List of
Solutions:
• If you are not restricted to a specific
interval and are asked to give the general
solutions then remember that adding on
any integer multiple of 2π represents a coterminal angle with the equivalent
trigonometric ratio.
Solutions:
7
2
k
6
x
11
2k
6
Where k is an integer and gives all
the coterminal angles of the
solution.
Practice
• Solve the equation. Find the general solutions
3 csc 2 0
3 csc 2
2
csc
3
3
which means that sin
2
2
2 k ,
2 k
3
3
Solving
Trigonometric Equations
Solve:
2 cos x 1 0
Step 1: Isosolate cos x using algebraic skills.
2 cos x 1
cos x 1
2
Step 2: Determine in which quadrants cosine is positive. Use the inverse
function to assist by finding the angle in Quad I first. Then use that angle
as the reference angle for the other quadrant(s).
QI
x
QIV
5
3
,
3
Note: cosine is positive in
Quad I and Quad IV.
Note: The reference angle is /3.
Solving
Trigonometric Equations
Solve:
Step 1:
tan x 1 0
2
tan x 1
2
tan x 1
tan x 1
2
Step 2:
Q1
x
Note: Since there is a , all four quadrants
hold a solution with /4 being the reference
angle.
QIV
QIII
QII
3 5 7
4
,
4
,
4
,
4
Solving
Trigonometric Equations
2
Solve: cot x cos x 2cot x
Step 1:
cot x cos2 x 2cot x 0
cot x cos 2 x 2 0
cot x 0 or cos2 x 2 0
cos2 x 2
cos2 x 2
cos x 2
Step 2:
x
3
2
,
2
x
Note: There is no solution here because 2
lies outside the range for cosine.
Solving
Trigonometric Equations
Try these:
Solution
3 7
,
4 4
1.
tan x 1 0
x
2.
sec x 4 0
x
3.
3tan x tan x
x 0,
2
3
2 4 5
3
,
3
,
5
6
,
6
3
,
, ,
3
7 11
,
6
6
Solving
Trigonometric Equations
Solve:
2sin 2 x sin x 1 0
2sin x 1sin x 1 0
Factor the quadratic equation.
2sin x 1 0 or sin x 1 0
Set each factor equal to zero.
1
sin x
2
7 11
x
,
6
6
sin x 1
x
2
Solve for sin x
Determine the correct quadrants
for the solution(s).
Solve : 4sin 2 ( x) 1 0 over the int erval [0, 2 )
This is a difference of squares and can factor
(2sin x 1)(2sin x 1) 0
Solve each factor and you should end up with 4 solutions
1
1
sin x
and sin x
2
2
x
5 7 11
6
,
6
,
6
,
6
Practice
Find the general solutions for
tan x 2 tan x 1
2
tan x 2 tan x 1 0
2
(tan x 1)(tan x 1) 0
tan x 1
3
7
x
k ,
k
4
4
Writing in terms of 1 trig fnc
• If there is more than one trig function
involved in the problem, then use your
identities.
• Replace one of the trig functions with an
identity so there is only one trig function
being used
Solve the following
2cos x sin x 1 0
2
Replace cos2 with 1 - sin2
2(1 sin 2 x) sin x 1 0
2 2sin 2 x sin x 1 0
2sin 2 x sin x 1 0
2sin 2 x sin x 1 0
(2sin x 1)(sin x 1) 0
1
sin x
and sin x 1
2
7
11
x
2k , x
2k , x 2k
6
6
2
Solving
Trigonometric Equations
Solve:
2sin 2 x 3cos x 3 0
2 1 cos 2 x 3cos x 3 0
2 2cos2 x 3cos x 3 0
2cos2 x 3cos x 1 0
2cos2 x 3cos x 1 0
2cos x 1cos x 1 0
2cos x 1 0 or cos x 1 0
cos x
x
1
2
5
3
,
3
Replace sin2x with 1-cos2x
Distribute
Combine like terms.
Multiply through by – 1.
Factor.
Set each factor equal to zero.
cos x 1
Solve for cos x.
x0
Determine the solution(s).
Solving
Trigonometric Equations
Solve:
cos x 1 sin x
cos x 1
2
sin x
Square both sides of the equation
in order to change sine into terms
of cosine giving only one trig
function to work with.
2
cos2 x 2cos x 1 sin 2 x
FOIL or Double Distribute
cos2 x 2cos x 1 1 cos2 x
2cos2 x 2cos x 0
2cos x cos x 1 0
2cos x 0 or cos x 1 0
cos x 1
cos x 0
x
Replace sin2x with 1 – cos2x
Set equation equal to zero since it is a
quadratic equation.
Factor
Set each factor equal to zero.
Solve for cos x
3
2
,
X2 x
Why is 3/2 removed as a solution?
Determine the solution(s).
It is removed because it does not
check in the original equation.
Solving
Trigonometric Equations
Solve:
Solution:
1
2
cos 3 x
No algebraic work needs to be done because cosine is already by itself.
Remember, 3x refers to an angle and one cannot divide by 3 because it
is cos 3x which equals ½.
Since 3x refers to an angle, find the angles whose cosine value is ½.
3x
x
5
3
,
Now divide by 3 because it is angle equaling angle.
3
5
9
3x
,
Notice the solutions do not exceed 2. Therefore,
more solutions may exist.
9
5 7 11
,
,
,
3
3 3
5 7 11
,
,
x ,
9
9 9 9
3
Return to the step where you have 3x equaling
the two angles and find coterminal angles for
those two.
Divide those two new angles by 3.
Solving
Trigonometric Equations
5 7 11 13 17
3x ,
,
,
,
,
3 3 3
3
3
3
x
The solutions still do not exceed 2.
Return to 3x and find two more
coterminal angles.
5 7 11 13 17
,
,
,
,
,
9 9
9
9
9
9
Divide those two new angles by 3.
5 7 11 13 17 19 23 The solutions still do not exceed 2.
3x ,
,
,
,
,
,
,
Return to 3x and find two more
3 3 3
3
3 coterminal angles.
3
3
3
x
5 7 11 13 17 19
,
,
,
,
,
,
9
9 9
9
9
9
9
Divide those two new angles by 3.
Notice that 19/9 now exceeds 2 and
is not part of the solution.
Therefore the solution to cos 3x = ½ is
x
5 7 11 13 17
9
,
9
,
9
,
9
,
9
,
9
Solving
Trigonometric Equations
Try these:
1.
4sin 2 x 2cos x 1
2.
csc x cot x 1
3.
3
sin 2 x
2
4.
x
2
cos
2
2
Solution
x 5.4218
x
2
2 5 5 11
x
,
,
,
3 6 3 6
x
2
Find the general solutions for
sin 3x +2= 1
sin 3x 1
3
3x
2
3
2k
2
x
3
3
2
x k
2 3
Practice
Solve 2cos 4x 3 0
3
cos 4 x
2
5
7
4x
and 4 x
6
6
5 k
7 k
x
, x
24 2
24 2