Precalculus Examples

$\sin (t + \frac{π}{6}) - \sin (t - \frac{π}{6})$

Step 1

Simplify each term.

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Step 1.1

Apply the sum of angles identity.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.3

Apply the difference of angles identity.

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

Step 1.4

Simplify each term.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- - \frac{\cos (t)}{2}$ .

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Step 1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.2

Multiply $\frac{\cos (t)}{2}$ by $1$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

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

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Step 2.2.1

Subtract $\sin (t) \sqrt{3}$ from $\sin (t) \sqrt{3}$ .

$\frac{0 + \cos (t) + \cos (t)}{2}$

Step 2.2.2

Add $0$ and $\cos (t)$ .

$\frac{\cos (t) + \cos (t)}{2}$

Step 2.3

Add $\cos (t)$ and $\cos (t)$ .

$\frac{2 \cos (t)}{2}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\frac{2 \cos (t)}{2}$

Step 2.4.2

Divide $\cos (t)$ by $1$ .

$\cos (t)$