Trigonometry Examples

$\sin (x + \frac{π}{4}) + \sin (x - \frac{π}{4})$

Step 1

Simplify each term.

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Step 1.1

Apply the sum of angles identity.

$\sin (x) \cos (\frac{π}{4}) + \cos (x) \sin (\frac{π}{4}) + \sin (x - \frac{π}{4})$

Step 1.2

Simplify each term.

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Step 1.2.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\sin (x) \frac{\sqrt{2}}{2} + \cos (x) \sin (\frac{π}{4}) + \sin (x - \frac{π}{4})$

Step 1.2.2

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

$\frac{\sin (x) \sqrt{2}}{2} + \cos (x) \sin (\frac{π}{4}) + \sin (x - \frac{π}{4})$

Step 1.2.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sin (x) \sqrt{2}}{2} + \cos (x) \frac{\sqrt{2}}{2} + \sin (x - \frac{π}{4})$

Step 1.2.4

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

$\frac{\sin (x) \sqrt{2}}{2} + \frac{\cos (x) \sqrt{2}}{2} + \sin (x - \frac{π}{4})$

Step 1.3

Apply the difference of angles identity.

$\frac{\sin (x) \sqrt{2}}{2} + \frac{\cos (x) \sqrt{2}}{2} + \sin (x) \cos (\frac{π}{4}) - \cos (x) \sin (\frac{π}{4})$

Step 1.4

Simplify each term.

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Step 1.4.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sin (x) \sqrt{2}}{2} + \frac{\cos (x) \sqrt{2}}{2} + \sin (x) \frac{\sqrt{2}}{2} - \cos (x) \sin (\frac{π}{4})$

Step 1.4.2

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

$\frac{\sin (x) \sqrt{2}}{2} + \frac{\cos (x) \sqrt{2}}{2} + \frac{\sin (x) \sqrt{2}}{2} - \cos (x) \sin (\frac{π}{4})$

Step 1.4.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.4.4

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Reorder the factors in the terms $\frac{\cos (x) \sqrt{2}}{2}$ and $- \frac{\sqrt{2} \cos (x)}{2}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

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

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Divide $\sin (x) \sqrt{2}$ by $1$ .

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