This document contains solutions to trigonometric equations on various intervals. It first determines that 4π/5 is not a solution to 2sin(2θ)=0. It then finds that 4π/5 and 7π/6 are solutions to 2cos(3θ)-5=0 on the interval [0,2π]. Similarly, it finds all solutions to equations involving tangent, cosine, sine and combinations of trigonometric functions on specified intervals through repetitive use of trigonometric identities and interval arithmetic.

1. Section 7.3
Trigonometric
Equations
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2. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

3. Determine whether is a solution of the equation 2sin 2 0.
4
5
Is a solution?
4
π
θ θ
π
θ
= + =
=
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2
2sin 2 2 2 2 2 2 2 0
4 2
π  
+ = + = + = ≠ ÷ ÷
 
is NOT a solution.
4
π
2
2sin 2 2 2 2 2
4
0
5
2
π  
+ = − + = − + = ÷ ÷
 
5
is a solution.
4
π

4. 2cos 3 0θ − =
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3
cos
2
θ =
[ )
3
Where is cos ? On the interval
2
0,2 there are two answers, one in
quadrant I and one in quadrant IV.
θ
π
=
[ )
11
On 0,2 , or =
6 6
π π
π θ θ=
A general formula would include all coterminal angles
11
2 or = 2
6 6
k k
π π
θ π θ π= + +

5. ( )Solve the equation: 2cos 2 1 0, 0 2θ θ π− = ≤ <
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( )
1
cos 2
2
θ = [ )
1
On the interval 0,2 the cosine of an angle is
2
5
when the angle is or .
3 3
π
π π
[ )
5
2 or 2 but since we divide by 2, adding 2 to the angle
3 3
will still give us a solution for that falls in the interval 0,2 .
π π
θ θ π
θ π
= =
5 5
2 , 2 +2 , 2 , 2 2
3 3 3 3
π π π π
θ θ π θ θ π= = = = +
5 7 11
, , ,
6 6 6 6
π π π π 
 
 

6. ( )Solve the equation: 3 tan 3 1 0, 0 2θ θ π+ = ≤ <
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( )
1
tan 3
3
θ = − [ )On the interval 0,2 the tangent of an angle is
3 5 11
when the angle is or .
3 6 6
π
π π
−
[ )
5 11
3 or 3 but since we divide by 3, adding 2 and 4 to
6 6
the angle will still give us a solution for that falls in the interval 0,2 .
π π
θ θ π π
θ π
= =
5 5 5 11 11 11
3 , 3 +2 , 3 +4 ,3 , 3 2 ,3 4
6 6 6 6 6 6
π π π π π π
θ θ π θ π θ θ π θ π= = = = = + = +
5 11 17 23 29 35
, , , , ,
18 18 18 18 18 18
π π π π π π 
 
 

7. Solve the equation: cos 1, 0 2
4
π
θ θ π
 
− = ≤ < ÷
 
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The cosine equals 1 when the angle is 0.
0
4
π
θ − =
4
π
θ =

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9. cos 0.2θ =
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1
cos 0.2 1.37θ −
= ≈
For inverse cosine the
calculator will only give an
angle from 0 to π. There is
another angle in the interval
from 0 to 2π with that cosine
value in quadrant IV.
−1
1
x
y
1.37θ =2 1.37θ π= −
{ }1.37,4.91

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11. 2
Solve the equation: 2cos cos 1 0, 0 2θ θ θ π− − = ≤ <
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( ) ( )2cos 1 cos 1 0θ θ+ − =
2cos 1 0 or cos 1 0θ θ+ = − =
1
cos or cos 1
2
θ θ == −
2 4
, , 0
3 3
π π
θ θ = θ= =
2 4
0, ,
3 3
π π 
 
 

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13. 2 2
Solve the equation: sin sin cos , 0 2θ θ θ θ π− = ≤ <
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2 2
sin sin 1 sinθ θ θ− = −
2
2sin sin 1 0θ θ− − =
( ) ( )2sin 1 sin 1 0θ θ+ − =
2sin 1 0 or sin 1 0θ θ+ = − =
1
sin or sin 1
2
θ θ= − =
7 11
, ,
6 6 2
π π π
θ θ = θ= =
7 11
, ,
2 6 6
π π π 
 
 

14. 2 2
Solve the equation: 3sin cos , 0 2θ θ θ π= ≤ <
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2 2
3sin 1 sinθ θ= − 2
4sin 1θ = 2 1
sin
4
θ =
1 1
sin
4 2
θ = ± = ±
θ =
π
6
,
5π
6
,
7π
6
,
11π
6
π
6
,
5π
6
,
7π
6
,
11π
6





