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What is x if sin^2x=1/4 ?

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gentzisisaci | Student, Grade 10 | eNoter

Posted December 21, 2010 at 1:02 AM via web

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What is x if sin^2x=1/4 ?

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giorgiana1976 | College Teacher | Valedictorian

Posted December 21, 2010 at 1:05 AM (Answer #1)

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We'll multiply by 4 both sides:

4(sin x)^2 = 1

We'll subtract 1 both sides:

4(sin x)^2 - 1 = 0

We'll re-write the difference of squares as a product:

4(sin x)^2 - 1 = (2 sin x - 1)(2 sin x + 1)

We'll put the product equal to zero:

(2 sin x - 1)(2 sin x + 1) = 0

We'll put each factor as zero:

2 sin x - 1 = 0

We'll add 1 both sides:

2 sin x = 1

We'll divide by 2:

sin x = 1/2

x = (-1)^k*arcsin (1/2) + k*pi

x = (-1)^k*pi/6 + k*pi

We'll put the other factor as zero:

2 sin x + 1 = 0

We'll subtract 1 both sides:

2 sin x = -1

sin x = -1/2

x = (-1)^(k+1)*pi/6 + k*pi

The solutions of the equation are: {(-1)^k*pi/6 + k*pi}U{(-1)^(k+1)*pi/6 + k*pi}.

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neela | High School Teacher | Valedictorian

Posted December 21, 2010 at 1:11 AM (Answer #3)

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What is x if sin^2x=1/4.

Sin^2x = 1/4.

=> sinx^2x- 1/4 = 0.

=> sin^x-(1/2)^2 = 0.

=> (sinx+1/2)(sinx-1/2) = 0, as (a^2-b^2 =(a+b)(a-b).

=> (sinx+1/2 = 0, or sinx-1/2 = 0.

sinx+1/2 = 0 gives sinx = -1/2 , or sinx = npi-(-1)^n*pi/6, for n= 0, 1,2,...

Sinx-1/2 = 0 gives x = npi+(-1)^n*pi/6, n= 0,1,2...

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