matematika - persamaan trigonometri sederhana
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Trigonometry SIMPLE
EQUATION
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Trigonomentri simple equation
is an equation that contains the
comparison trigonomentri
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In general, to solve trigonometry is used in the
following formula:
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1. sin x = sin α
x = α + k.360⁰ atau x = (180⁰ - α) + k.360⁰
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2. cos x = cos α
x = α + k.360⁰ atau x = - α + k.360⁰
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3. tan x = tan α
x = α + k.180⁰
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For angle in units of radians, in use the following formula:
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1. sin x = sin α
x = α + k.2π or x = (π – α) + k.2π
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2. cos x = sin α
x = α + k.2π or x = -α + k.2π
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3. tan x = tan α
x = α + k.π
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Example - example problems trigonometric
equations:
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Determine the set of equations following the completion of the interval 0 ≤ x ≤ 2π
a. Sin x = ½ √3
b. Tan x = √3
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Answer :
a. Sin x = ½√3
= sin (π/3 + k. 2π)
x = π/3 + k. 2π
for k = 0 x = π/3
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OR
sin x = ½ √3
= sin ( π – π/3 + k. 2π)
x = 2π/3 + k . 2π
For k = 0 x = 2π/3
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Thus, the solution set
= {π / 3, 2π / 3}
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tan x = √3
= tan (π/3 + k. π )
x = π/3 + k . π
For k = 0 x = π/3 + k . Π
k = 1 x = 4π/3
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so, the solution set
= {Π / 3, 4π / 3}
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2. Determine the set of completion of the equation cos (3x - 45 ⁰) = - ½ √ 2, for 0 ⁰ ≤ x ≤ 360 ⁰.
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Answer :
Cos (3x – 45⁰) = -½√2
Cos (3x – 45⁰) = cos 135⁰
3x – 45⁰ = 135⁰ + k. 360⁰
3x = 180⁰ + k. 360⁰
x = 60⁰ + k. 120⁰
x = 60⁰ , 180⁰ , 300⁰
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OR
3x – 45⁰ = -135 + k . 360⁰
3x = -90⁰ + k . 360⁰
x = -30⁰ + k . 120⁰
x = 90⁰ , 210⁰ , 330⁰
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Thus, the solution set = {60 ⁰, 90 ⁰, 180 ⁰, 210 ⁰, 300 ⁰, 330 ⁰}
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The basic technique is to solve
trigonometric equations using trig identities and algebra techniques to transform a trigonometric equation into simpler forms.
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example:
Determine the set of completion of sin x = sin 70°, 0° <x <360 °
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Answer :
x = 70° + k.360° k = 0 ==> x = 70° atau
x = (180 - 70) + k.360° ==> x = 110° + k.360°
k = 0 ==> x = 110°
Jadi, Hp = {70°, 110°}
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Determine the set of completion of cos x = cos 24 in the interval 0 ° <x <360 °
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Answer :
x = 24° + k.360° k = 0 , x = 24° OR
x = -24° + k.360°
k = 1 , x = -24° + 360° = 336°
Thus, Hp = {24°, 336°}
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Determine the set of completion of tan x = tan 56 °, in the interval 0 ° <x
<360 °
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Answer:
x = 56 ° + ° k.180
k = 0 ==> x = 56 °
k = 1 ==> x = 56 ° + 180 ° = 236 °
Thus, the solution set is
{52 °, 236 °}
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THANK YOU