find the general solution of sin 3 theeta=0
Answers
Answer:
Let us begin with a basic equation, sin x = 0. The principal solution for this case will be x = 0,π,2π as these values satisfy the given equation lying in the interval [0, 2π] . But, we know that if sin x = 0, then x = 0, π, 2π, π, -2π, -6π, etc. are solutions of the given equation. Hence, the general solution for sin x = 0 will be, x = nπ, where n∈I.
Similarly, general solution for cos x = 0 will be x = (2n+1)π/2, n∈I, as cos x has a value equal to 0 at π/2, 3π/2, 5π/2, -7π/2, -11π/2 etc. Below here is the table defining the general solutions of the given trigonometric functions involved equations.
EquationsSolutionssin x = 0 x = nπcos x = 0x = (nπ + π/2)tan x = 0x = nπsin x = 1x = (2nπ + π/2) = (4n+1)π/2cos x = 1x = 2nπsin x = sin θx = nπ + (-1)nθ, where θ ∈ [-π/2, π/2]cos x = cos θx = 2nπ ± θ, where θ ∈ (0, π]tan x = tan θx = nπ + θ, where θ ∈ (-π/2 , π/2]sin2 x = sin2 θx = nπ ± θcos2 x = cos2 θx = nπ ± θtan2 x = tan2 θx = nπ ± θ
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Answer:
There is another way to solve sin 3x = 0.
Use the trig identity:
sin
3
x
=
3
sin
x
−
4
sin
3
x
sin
3
x
=
sin
x
(
3
−
4
sin
2
x
)
a. sin x = 0 --> x = 0, and
x
=
π
, and
x
=
2
π
.
b.
(
3
−
4
sin
2
x
)
=
0
4
sin
2
x
=
3
sin
2
x
=
3
4
-->
sin
x
=
±
√
3
2
- When
sin
x
=
√
3
2
-->
x
=
π
3
and
x
=
2
π
3
- When
sin
x
=
−
√
3
2
-->
x
=
−
π
3
and
x
=
−
2
π
3
HOPE THIS HELPS U
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