Trigonometry Examples

$\sin (\frac{11 x}{2}) + \sin (\frac{7 x}{2})$

Step 1

Simplify each term.

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Step 1.1

Apply the sine half-angle identity.

$\pm \sqrt{\frac{1 - \cos (11 x)}{2}} + \sin (\frac{7 x}{2})$

Step 1.2

Rewrite $\sqrt{\frac{1 - \cos (11 x)}{2}}$ as $\frac{\sqrt{1 - \cos (11 x)}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (11 x)}}{\sqrt{2}} + \sin (\frac{7 x}{2})$

Step 1.3

Multiply $\frac{\sqrt{1 - \cos (11 x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (11 x)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \sin (\frac{7 x}{2})$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{\sqrt{1 - \cos (11 x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{\sqrt{2} \sqrt{2}} + \sin (\frac{7 x}{2})$

Step 1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sin (\frac{7 x}{2})$

Step 1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sin (\frac{7 x}{2})$

Step 1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sin (\frac{7 x}{2})$

Step 1.4.5

Add $1$ and $1$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{{\sqrt{2}}^{2}} + \sin (\frac{7 x}{2})$

Step 1.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sin (\frac{7 x}{2})$

Step 1.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sin (\frac{7 x}{2})$

Step 1.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{2^{\frac{2}{2}}} + \sin (\frac{7 x}{2})$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{2^{\frac{2}{2}}} + \sin (\frac{7 x}{2})$

Step 1.4.6.4.2

Rewrite the expression.

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{2^{1}} + \sin (\frac{7 x}{2})$

Step 1.4.6.5

Evaluate the exponent.

$\pm \frac{\sqrt{1 - \cos (11 x)} \sqrt{2}}{2} + \sin (\frac{7 x}{2})$

Step 1.5

Combine using the product rule for radicals.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} + \sin (\frac{7 x}{2})$

Step 1.6

Apply the sine half-angle identity.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \sqrt{\frac{1 - \cos (7 x)}{2}}$

Step 1.7

Rewrite $\sqrt{\frac{1 - \cos (7 x)}{2}}$ as $\frac{\sqrt{1 - \cos (7 x)}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)}}{\sqrt{2}}$

Step 1.8

Multiply $\frac{\sqrt{1 - \cos (7 x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.9

Combine and simplify the denominator.

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Step 1.9.1

Multiply $\frac{\sqrt{1 - \cos (7 x)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.9.2

Raise $\sqrt{2}$ to the power of $1$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.9.3

Raise $\sqrt{2}$ to the power of $1$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.9.5

Add $1$ and $1$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.9.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.9.6.4

Cancel the common factor of $2$ .

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Step 1.9.6.4.1

Cancel the common factor.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.9.6.4.2

Rewrite the expression.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{2^{1}}$

Step 1.9.6.5

Evaluate the exponent.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{1 - \cos (7 x)} \sqrt{2}}{2}$

Step 1.10

Combine using the product rule for radicals.

$\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{(1 - \cos (7 x)) \cdot 2}}{2}$

Step 2

Reorder factors in $\pm \frac{\sqrt{(1 - \cos (11 x)) \cdot 2}}{2} \pm \frac{\sqrt{(1 - \cos (7 x)) \cdot 2}}{2}$ .

$\pm \frac{\sqrt{2 (1 - \cos (11 x))}}{2} \pm \frac{\sqrt{2 (1 - \cos (7 x))}}{2}$