Solving Trig Equations using
Double and Half Angle
Formulas
Lori Jordan
Kate Dirga
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Printed: June 21, 2016
AUTHORS
Lori Jordan
Kate Dirga
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Chapter 1. Solving Trig Equations using Double and Half Angle Formulas
C HAPTER
1
Solving Trig Equations
using Double and Half Angle Formulas
Here you’ll solve trig equations using the half and double angle formulas.
Trig Riddle #4: I am an angle x such that 0 ≤ x < 2π. I satisfy the equation sin 2x − sin x = 0. What angle am I?
Solve Trigonometric Equations
Lastly, we can use the half and double angle formulas to solve trigonometric equations.
Solve the following trigonometric equations
Solve tan 2x + tan x = 0 when 0 ≤ x < 2π.
Change tan 2x and simplify.
tan 2x + tan x = 0
2 tan x
+ tan x = 0
1 − tan2 x
2 tan x + tan x(1 − tan2 x) = 0
→ Multiply everything by 1 − tan2 x to eliminate denominator.
2 tan x + tan x − tan3 x = 0
3 tan x − tan3 x = 0
tan x(3 − tan2 x) = 0
Set each factor equal to zero and solve.
3 − tan2 x = 0
− tan2 x = −3
tan x = 0
x = 0 and π
and
tan2 x = 3
√
tan x = ± 3
π 2π 4π 5π
x= , , ,
3 3 3 3
Solve 2 cos 2x + 1 = 0 when 0 ≤ x < 2π.
In this case, you do not have to use the half-angle formula. Solve for 2x .
x
2 cos + 1 = 0
2
x
2 cos = −1
2
x
1
cos = −
2
2
1
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Now, let’s find cos a = − 12 and then solve for x by dividing by 2.
x 2π 4π
= ,
2
3 3
4π 8π
= ,
3 3
Now, the second solution is not in our range, so the only solution is x =
Solve 4 sin x cos x =
4π
3 .
√
3 for 0 ≤ x < 2π.
Pull a 2 out of the left-hand side and use the sin 2x formula.
4 sin x cos x =
2 · 2 sin x cos x =
2 · sin 2x =
sin 2x =
2x =
x=
√
3
√
3
√
3
√
3
2
π 5π 7π 11π
, , ,
3 3 3 3
π 5π 7π 11π
, , ,
6 6 6 6
Examples
Example 1
Earlier, you were asked what is the angle.
Use the double angle formula and simplify.
sin 2x − sin x = 0
2 sin x cos x − sin x = 0
sin x(2 cos x − 1) = 0
1
sin x = 0OR cos x =
2
Under the constraint 0 ≤ x < 2π, sin x = 0 when x = 0 or when x = π. Under this same constraint, cos x =
x = π3 or when x = 5π
3 .
Example 2
Solve the following equation for 0 ≤ x < 2π.
sin 2x = −1
2
1
2
when
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Chapter 1. Solving Trig Equations using Double and Half Angle Formulas
sin
x
= −1
2
x 3π
=
2
2
x = 3π
From this we can see that there are no solutions within our interval.
Example 3
Solve the following equation for 0 ≤ x < 2π.
cos 2x − cos x = 0
cos 2x − cos x = 0
2
2 cos x − cos x − 1 = 0
(2 cos x − 1)(cos x + 1) = 0
Set each factor equal to zero and solve.
2 cos x − 1 = 0
2 cos x = 1
1
cos x =
2
π 5π
x= ,
3 3
cos x + 1 = 0
and
cos x = −1
x=π
Review
Solve the following equations for 0 ≤ x < 2π.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
cos x − cos 12 x = 0
sin 2x cos x = sin x
cos 3x − cos3 x = 3 sin2 x cos x
tan 2x − tan x = 0
cos 2x − cos x = 0
2 cos2 2x = 1
tan 2x = 4
cos 2x = 1 + cos x
sin 2x + sin x = 0
cos2 x − cos 2x = 0
cos 2x
=1
cos2 x
cos 2x − 1 = sin2 x
cos 2x = cos x
sin 2x − cos 2x = 1
sin2 x − 2 = cos 2x
cot x + tan x = 2 csc 2x
3
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Answers for Review Problems
To see the Review answers, open this PDF file and look for section 14.17.
4