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Algebraically solve the exact values of all angles in the interval [0,pi] that satisfy the equation: 2sin^2(x) = 1 - cos(x)

Sagot :

Given equation is

[tex]2\sin ^2(x)=1-\cos (x)[/tex][tex]\text{ Use }\sin ^2(x)=1-\cos ^2(x).[/tex]

[tex]2(1-\cos ^2(x))=1-\cos (x)[/tex]

[tex]2(1^2-\cos ^2(x))=1-\cos (x)[/tex][tex]\text{ Use }1^2-\cos ^2(x)=(1-\cos (x))(1+\cos (x)).[/tex][tex]2(1-\cos (x))(1+\cos (x))=1-\cos (x)[/tex]

Cancel out the common term on both sides, we get

[tex]2(1+\cos (x))=1[/tex]

Dividing both sides by 2, we get

[tex]1+\cos (x)=\frac{1}{2}[/tex]

Subtracting 1 from both sides, we get

[tex]1+\cos (x)-1=\frac{1}{2}-1[/tex]

[tex]\cos (x)=-\frac{1}{2}[/tex]

[tex]\text{Use }\cos (\frac{2\pi}{3})=-\frac{1}{2}\text{.}[/tex]

[tex]\cos (x)=\cos (\frac{2\pi}{3})[/tex][tex]x=\frac{2\pi}{3}[/tex]

Hence the exact value of x is

[tex]x=\frac{2\pi}{3}[/tex]

The exact values of all angles in the interval [0, pi] that satisfy the given equation is 2pi/3.