Algebra Examples

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

Step 1

Start on the left side.

$\sin (\frac{π}{6} - x)$

Step 2

Apply the difference of angles identity.

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Reorder factors in $\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2}$ .

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

Step 5

Now consider the right side of the equation.

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

Combine $\frac{1}{2}$ and $\sqrt{3}$ .

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

Step 6.3.2

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

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

Step 6.4

Reorder factors in $\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2}$ .

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

Step 7

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