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

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

Step 4

Now consider the right side of the equation.

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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