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MTH 113 Section 5.5
Trigonometric Equations
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Equations Involving A Single Trigonometric Function
Solve cos x =
2
2
x=
π
4
is one solution
There are two solution between 0 and 2π.
They are
π
4
and
7π
.
4
But, any angle that is co‐terminal with π/4 or 7π/4 has cosine equal to 2
2
All solutions to this equation are:
x=
π
4
+ 2nπ and x =
7π
+ 2nπ , where n is any integer.
4
Find all solutions of each equation.
cos x =
3
2
2sin x + 3 = 0
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5sin θ + 1 = 3sin θ
7 cos θ + 9 = −2 cos θ
Solve for x where 0 ≤ x<2π :
3 sin x = 3 + sin x
Solve for x where 0 ≤ x<2π :
sin x + cos x = 0
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Solve for x where 0 ≤ x<2π :
sin x tan x = sin x
Equations Involving Multiple Angles
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Solve for x: tan 2x= 3, 0 ≤ x<2π
Solve each equation on the interval [0, 2π].
cos 2 x =
2
2
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tan
x
3
=
2
3
cos
2θ
= −1
3
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π⎞
2
⎛
sin ⎜ 2 x − ⎟ =
4⎠ 2
⎝
Trigonometric Equations Quadratic in Form
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2sin 2 x + sin x − 1 = 0
cos 2 + 2 cos x − 3 = 0
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cos 2 − 1 = 0
3 tan 2 x − 9 = 0
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4 sec 2 x − 2 = 0
(tan x + 1)(sin x − 1) = 0
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cot x(tan x + 1) = 0
cos x − 2sin x cos x = 0
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Use an identity to solve each equation on the interval [0, 2π].
2 cos 2 x − sin x − 1 = 0
3cos 2 x = sin 2 x
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sin 2 x = sin x
cos 2 x + cos x + 1 = 0
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sin x + cos x = −1
π⎞
π⎞
⎛
⎛
sin ⎜ x + ⎟ + sin ⎜ x − ⎟ = 1
3⎠
3⎠
⎝
⎝
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Use a calculator to solve correct to four decimal places in[0,2π].
Use radian measure.
sinx=0.7392
3cos 2 x − 8cos x − 3 = 0
2sin2x + 3sinx – 4 = 0
15