5.3 SOLVING TRIG EQUATIONS
SOLVING TRIG EQUATIONS

Solve the following equation for x:
Sin x = ½
SOLVING TRIG EQUATIONS




In this section, we will be solving various types of
trig equations
You will need to use all the procedures learned
last year in Algebra II
All of your answers should be angles.
Note the difference between finding all solutions
and finding all solutions in the domain [0, 2π)
SOLVING TRIG EQUATIONS

Guidelines to solving trig equations:
1)
Isolate the trig function
2)
Find the reference angle
3)
Put the reference angle in the proper quadrant(s)
4)
Create a formula for all possible answers (if
necessary)
SOLVING TRIG EQUATIONS
1- 2 Cos x = 0
1) Isolate the trig function
1- 2 Cos x = 0
+ 2 Cos x = + 2 Cos x
1= 2 Cos x
2
2
Cos x = ½
SOLVING TRIG EQUATIONS
Cos x = ½
2) Find the reference angle

x=
3
3) Put the reference angle in the proper quadrant(s)

5
I=
IV =
3
3
SOLVING TRIG EQUATIONS
Cos x = ½
4) Create a formula if necessary

x =  2n
3
5
 2n
x=
3
SOLVING TRIG EQUATIONS

Find all solutions to the following equation:
Sin x + 1 = - Sin x
+ Sin x
+ Sin x
→ 2 Sin x + 1 = 0
-1 -1
→ 2 Sin x = -1
→ Sin x = - ½
SOLVING TRIG EQUATIONS
Sin x = - ½

Ref. Angle:
6
7
 2n
III:
6
11
 2n
Iv:
6
Quad.: III, IV
SOLVING TRIG EQUATIONS

Find the solutions in the interval [0, 2π) for the
following equation:
Tan²x – 3 = 0
Tan²x = 3
Tan x =  3
SOLVING TRIG EQUATIONS
Tan x =  3

Ref. Angle:
3

I:
3
2
II:
3
Quad.: I, II, III, IV
4
III:
3
 2 4 5
x= 3, 3 , 3 , 3
5
IV:
3
SOLVING TRIG EQUATIONS
 Solve
the following equations for all real
values of x.
a)
Sin x + 2 = - Sin x
b)
3Tan² x – 1 = 0
c)
Cot x Cos² x = 2 Cot x
SOLVING TRIG EQUATIONS

Find all solutions to the following equation:
Sin x + 2 = - Sin x
2 Sin x = - 2
2
Sin x = 2
5
x = 4  2n
7
x = 4  2n
SOLVING TRIG EQUATIONS
3Tan² x – 1 = 0
1
Tan² x =
3
1
Tan x = 
3

x = 6  2n
5
 2n
x=
6
7
 2n
x=
6
11
 2n
x=
6
SOLVING TRIG EQUATIONS
Cot x Cos² x = 2 Cot x
Cot x Cos² x – 2 Cot x = 0
Cot x (Cos² x – 2) = 0
Cot x = 0
Cos² x – 2 = 0
Cos x = 0
Cos² x – 2 = 0
x =  2n
Cos x =  2
2
3
 2n
x=
2
No Solution
SOLVING TRIG EQUATIONS
 

 
2  
5.3 SOLVING TRIG EQUATIONS
SOLVING TRIG EQUATIONS
 Find
all solutions to the following equation.
4 Tan²x – 4 = 0
Tan²x = 1
Tan x = ±1 
Ref. Angle =
4

x = 4  n
3
 n
x=
4
SOLVING TRIG EQUATIONS

Equations of the Quadratic Type

Many trig equations are of the quadratic type:
2Sin²x – Sin x – 1 = 0
 2Cos²x + 3Sin x – 3 = 0


To solve such equations, factor the quadratic or,
if that is not possible, use the quadratic formula
SOLVING TRIG EQUATIONS
 Solve
the following on the interval [0, 2π)
2Cos²x + Cos x – 1 = 0 2x² + x - 1
If possible, factor the equation into two binomials.
(2Cos x – 1) (Cos x + 1) = 0
Now set each factor equal to zero
SOLVING TRIG EQUATIONS
2Cos x – 1 = 0
Cos x = ½

Ref. Angle:
3
Quad: I, IV
 5
x= ,
3 3
Cos x + 1 = 0
Cos x = -1
x= 
SOLVING TRIG EQUATIONS
 Solve
the following on the interval [0, 2π)
2Sin²x - Sin x – 1 = 0
(2Sin x + 1) (Sin x - 1) = 0
SOLVING TRIG EQUATIONS
2Sin x + 1 = 0
Sin x = - ½

Ref. Angle:
6
Quad: III, IV
7 11
x= ,
6
6
Sin x - 1 = 0
Sin x = 1

x=
2
SOLVING TRIG EQUATIONS
 Solve
the following on the interval [0, 2π)
2Cos²x + 3Sin x – 3 = 0
Convert all expressions to one trig function
2 (1 – Sin²x) + 3Sin x – 3 = 0
2 – 2Sin²x + 3Sin x – 3 = 0
0 = 2Sin²x – 3Sin x + 1
SOLVING TRIG EQUATIONS
0 = 2Sin²x – 3Sin x + 1 0 = (2Sin x – 1) (Sin x – 1)
2Sin x - 1 = 0
Sin x = ½

Ref. Angle:
6
Quad: I, II
 5
x= ,
6
6
Sin x - 1 = 0
Sin x = 1

x=
2
SOLVING TRIG EQUATIONS
 Solve
the following on the interval [0, 2π)
2Sin²x + 3Cos x – 3 = 0
Convert all expressions to one trig function
2 (1 – Cos²x) + 3Cos x – 3 = 0
2 – 2Cos²x + 3Cos x – 3 = 0
0 = 2Cos²x – 3Cos x + 1
SOLVING TRIG EQUATIONS
0 = 2Cos²x – 3Cos x + 1 0 = (2Cos x – 1) (Cos x – 1)
2Cos x - 1 = 0
Cos x = ½

Ref. Angle:
3
Quad: I, IV
 5
x= ,
3
3
Cos x - 1 = 0
Cos x = 1
x= 0
SOLVING TRIG EQUATIONS

The last type of quadratic equation would be a
problem such as:
(
Sec x + 1 )² =
Tan²x
What do these two trig functions have in common?
When you have two trig functions that are related
through a Pythagorean Identity, you can square
both sides.
SOLVING TRIG EQUATIONS
(Sec x + 1)² = Tan²x
Sec²x + 2Sec x + 1 = Sec²x - 1
2 Sec x + 1 = -1
Sec x = -1
Cos x = -1
x= 
When you have a problem that requires you to
square both sides, you must check your answer
when you are done!
SOLVING TRIG EQUATIONS
Sec x + 1 = Tan x
1  1  0
x= 
SOLVING TRIG EQUATIONS
(Cos
Cosx x+ +1)²
1 = Sin
Sin²xx
Cos²x + 2Cos x + 1 = 1 – Cos² x
2Cos² x + 2 Cos x = 0
Cos x (2 Cos x + 2) = 0
Cos x = 0
Cos x = - 1
 3
x= ,
x= 
2
2
SOLVING TRIG EQUATIONS
Cos x + 1 = Sin x
 3
x= , , 
2 2
Cos

 1  Sin
2
0 11

2
3
3
Cos
 1  Sin
2
2
0  1  -1
Cos   1  Sin 
-1  1  0
5.3 SOLVING TRIG EQUATIONS
SOLVING TRIG EQUATIONS



Equations involving multiply angles
Solve the equation for the angle as your normally
would
Then divide by the leading coefficient
SOLVING TRIG EQUATIONS

Solve the following trig equation for all values of
x.
2Sin 2x + 1 = 0
2Sin 2x = -1
Sin 2x = - ½
7
 2n 
2x =
6
7
 n
x=
12
11
 2n
2x =
6
11
 n
x=
12
SOLVING TRIG EQUATIONS
x
x
3 T an    3  0  T an   - 1
2
2
x
3

 n
2
4
x
7

 n
2
4
x 3

 2n 
2
2
x
7

 2n 
2
2
Redundant
Answer
SOLVING TRIG EQUATIONS

Solve the following equations for all values of x.
a)
2Cos 3x – 1 = 0
b)
Cot (x/2) + 1 = 0
SOLVING TRIG EQUATIONS
2Cos 3x - 1 = 0
2Cos 3x = 1
Cos 3x = ½
3x =

3

 2n 
2n
x= 
9
3
5
 2n 
3x =
3
5 2n 

x=
9
3
SOLVING TRIG EQUATIONS
x
Cot    1  0
2
x
3

 n
2
4
x 3

 2n 
2
2
x
 Cot    - 1
2
SOLVING TRIG EQUATIONS

Topics covered in this section:

Solving basic trig equations
Finding solutions in [0, 2π)
 Find all solutions





Solving quadratic equations
Squaring both sides and solving
Solving multiple angle equations
Using inverse functions to generate answers
SOLVING TRIG EQUATIONS
Find all solutions to the following equation:
Sec²x – 3Sec x – 10 = 0
(Sec x + 2) (Sec x – 5) = 0
Sec x + 2 = 0
Sec x = -2
Cos x = - ½
2
 2n 
x=
3
4
 2n 
3
Sec x – 5 = 0
Sec x = 5
1
Cos x =
5
1
x = Cos  
5
-1
SOLVING TRIG EQUATIONS

a)
b)
c)
One of the following equations has solutions and
the other two do not. Which equations do not
have solutions.
Sin²x – 5Sin x + 6 = 0
Sin²x – 4Sin x + 6 = 0
Sin²x – 5Sin x – 6 = 0
Find conditions involving constants b and c that
will guarantee the equation Sin²x + bSin x + c = 0
has at least one solution.
SOLVING TRIG FUNCTIONS

Find all solutions of the following equation in the
interval [0, 2π)
Sec²x – 2 Tan x = 4
1 + Tan²x – 2Tan x – 4 = 0
Tan²x – 2Tan x – 3 = 0
(Tan x + 1) (Tan x – 3) = 0
Tan x = -1
Tan x = 3
SOLVING TRIG FUNCTIONS
Tan x = -1
3 7
X
,
4
4
Tan x = 3
x = ArcTan 3
ref. angle: 71.6º
Quad: I, III
x = 71.6º, 251.6º