Find all solutions in the interval [0, 2π).. . 4 sin2x - 4 sin x + 1 = 0 . . pi/6, 5pi/6. . pi/6, 11pi/6. . pi/3, 5pi/3. . 7pi/6, 11pi/6

We have the equation:4 sin^2 x - 4 sin x + 1 = 0For the substitution: t = sin x4 t^2 - 4 t + 1 = 0( 2 t - 1 )^2 = 02 t - 1 = 02 t = 1 /:2t = 1/2Therefore : sin x = 1/2 and x is in the interval [ 0, 2 π ).Answer: A ) x = π/6 and x = 5 π/6

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ardni313 ardni313 · Answer 2 · 2019-09-11T17:07:10+02:00

The value of x that satisfies equation 4 sin²x - 4 sin x + 1 = 0 is π/ 6 and 5/6π

Further explanation

Trigonometry is the science of mathematics that studies the relationship between sides and angles in triangles

Trigonometric identity is a sentence that contains trigonometric functions. The truth of this sentence is an identity that needs to be proven

There are several types of identity formulas that can be used starting from the basic formula of the Pythagoras theorem

x² + y² = r²

This declining formula includes:

cos²a + sin²a = 1

cot²a + 1 = csc²a

1 + tan²a = sec²a

From the above equation: 4 sin²x - 4 sin x + 1 = 0, we can say sin x = a, then the equation becomes:

4a² - 4a + 1 = 0

we divide 4 :

a² - a + 1/4

then :

(a-1/2) (a-1/2) = 0

a = 1/2

so that

sin x = 1/2

Because the interval between 0 - 2π and positive sin values are in quadrants 1 and 2 so the x values that meet are 30° and 150° or π/6 and 5/6 π

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Keywords: trigonometric identities, sin, cos