Verify the Identity sin(pi/6+x)=1/2(cos(x)+ square root of 3sin(x))

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Trigonometry

$\sin (\frac{π}{6} + x) = \frac{1}{2} (\cos (x) + \sqrt{3} \sin (x))$

Start on the left side.

$\sin (\frac{π}{6} + x)$

Apply the sum of angles identity.

$\sin (\frac{π}{6}) \cos (x) + \cos (\frac{π}{6}) \sin (x)$

Simplify each term.

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The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1}{2} \cos (x) + \cos (\frac{π}{6}) \sin (x)$

Combine $\frac{1}{2}$ and $\cos (x)$ .

$\frac{\cos (x)}{2} + \cos (\frac{π}{6}) \sin (x)$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\cos (x)}{2} + \frac{\sqrt{3}}{2} \sin (x)$

Combine $\frac{\sqrt{3}}{2}$ and $\sin (x)$ .

$\frac{\cos (x)}{2} + \frac{\sqrt{3} \sin (x)}{2}$

Now consider the right side of the equation.

$\frac{1}{2} \cdot (\cos (x) + \sqrt{3} \sin (x))$

Simplify.

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Apply the distributive property.

$\frac{1}{2} \cos (x) + \frac{1}{2} (\sqrt{3} \sin (x))$

Combine $\frac{1}{2}$ and $\cos (x)$ .

$\frac{\cos (x)}{2} + \frac{1}{2} (\sqrt{3} \sin (x))$

Multiply $\frac{1}{2} (\sqrt{3} \sin (x))$ .

$\frac{\cos (x)}{2} + \frac{\sqrt{3} \sin (x)}{2}$

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (\frac{π}{6} + x) = \frac{1}{2} \cdot (\cos (x) + \sqrt{3} \sin (x))$ is an identity

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