Precalculus Examples

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

Step 1

Start on the left side.

$\sin (x + \frac{π}{6}) - \cos (x + \frac{π}{3})$

Step 2

Apply the sum of angles identity.

$\sin (x) \cos (\frac{π}{6}) + \cos (x) \sin (\frac{π}{6}) - \cos (x + \frac{π}{3})$

Step 3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\sin (x) \cos (\frac{π}{6}) + \cos (x) \sin (\frac{π}{6}) - (\cos (x) \cos (\frac{π}{3}) - \sin (x) \sin (\frac{π}{3}))$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

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

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

Step 4.1.6

Apply the distributive property.

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

Step 4.1.7

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

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.7.2

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

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

Step 4.2

Combine the opposite terms in $\frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} - \frac{\cos (x)}{2} + \frac{\sqrt{3} \sin (x)}{2}$ .

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Step 4.2.1

Subtract $\frac{\cos (x)}{2}$ from $\frac{\cos (x)}{2}$ .

$\frac{\sin (x) \sqrt{3}}{2} + 0 + \frac{\sqrt{3} \sin (x)}{2}$

Step 4.2.2

Add $\frac{\sin (x) \sqrt{3}}{2}$ and $0$ .

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Reorder the factors of $\sin (x) \sqrt{3}$ .

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

Step 4.5

Add $\sqrt{3} \sin (x)$ and $\sqrt{3} \sin (x)$ .

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

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Divide $\sqrt{3} \sin (x)$ by $1$ .

$\sqrt{3} \sin (x)$

Step 5

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

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