Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

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

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

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 $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

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

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

Step 4.1.6

Apply the distributive property.

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

Step 4.1.7

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

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.7.2

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

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

Step 4.2

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

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Step 4.2.1

Subtract $\frac{\sqrt{3} \cos (x)}{2}$ from $\frac{\sqrt{3} \cos (x)}{2}$ .

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

Step 4.2.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Add $\sin (x)$ and $\sin (x)$ .

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

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Divide $\sin (x)$ by $1$ .

$\sin (x)$

Step 5

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

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