Trigonometry Examples

$11 = 5 + 4 \sin (\frac{π}{12}) (x + 10)$

Step 1

Rewrite the equation as $5 + 4 \sin (\frac{π}{12}) (x + 10) = 11$ .

$5 + 4 \sin (\frac{π}{12}) (x + 10) = 11$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

The exact value of $\sin (\frac{π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1.1.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$5 + 4 \sin (\frac{π}{4} - \frac{π}{6}) (x + 10) = 11$

Step 2.1.1.2

Apply the difference of angles identity.

$5 + 4 (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) (x + 10) = 11$

Step 2.1.1.3

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

$5 + 4 (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})) (x + 10) = 11$

Step 2.1.1.4

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

$5 + 4 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})) (x + 10) = 11$

Step 2.1.1.5

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

$5 + 4 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})) (x + 10) = 11$

Step 2.1.1.6

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

$5 + 4 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) (x + 10) = 11$

Step 2.1.1.7

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.1.1.7.1

Simplify each term.

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Step 2.1.1.7.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 2.1.1.7.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$5 + 4 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) (x + 10) = 11$

Step 2.1.1.7.1.1.2

Combine using the product rule for radicals.

$5 + 4 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) (x + 10) = 11$

Step 2.1.1.7.1.1.3

Multiply $2$ by $3$ .

$5 + 4 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) (x + 10) = 11$

Step 2.1.1.7.1.1.4

Multiply $2$ by $2$ .

$5 + 4 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) (x + 10) = 11$

Step 2.1.1.7.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.1.1.7.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$5 + 4 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) (x + 10) = 11$

Step 2.1.1.7.1.2.2

Multiply $2$ by $2$ .

$5 + 4 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) (x + 10) = 11$

Step 2.1.1.7.2

Combine the numerators over the common denominator.

$5 + 4 \frac{\sqrt{6} - \sqrt{2}}{4} (x + 10) = 11$

Step 2.1.2

Cancel the common factor of $4$ .

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Step 2.1.2.1

Cancel the common factor.

$5 + 4 \frac{\sqrt{6} - \sqrt{2}}{4} (x + 10) = 11$

Step 2.1.2.2

Rewrite the expression.

$5 + (\sqrt{6} - \sqrt{2}) (x + 10) = 11$

Step 2.1.3

Expand $(\sqrt{6} - \sqrt{2}) (x + 10)$ using the FOIL Method.

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Step 2.1.3.1

Apply the distributive property.

$5 + \sqrt{6} (x + 10) - \sqrt{2} (x + 10) = 11$

Step 2.1.3.2

Apply the distributive property.

$5 + \sqrt{6} x + \sqrt{6} \cdot 10 - \sqrt{2} (x + 10) = 11$

Step 2.1.3.3

Apply the distributive property.

$5 + \sqrt{6} x + \sqrt{6} \cdot 10 - \sqrt{2} x - \sqrt{2} \cdot 10 = 11$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

Move $10$ to the left of $\sqrt{6}$ .

$5 + \sqrt{6} x + 10 \cdot \sqrt{6} - \sqrt{2} x - \sqrt{2} \cdot 10 = 11$

Step 2.1.4.2

Multiply $10$ by $- 1$ .

$5 + \sqrt{6} x + 10 \sqrt{6} - \sqrt{2} x - 10 \sqrt{2} = 11$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $5$ from both sides of the equation.

$\sqrt{6} x + 10 \sqrt{6} - \sqrt{2} x - 10 \sqrt{2} = 11 - 5$

Step 3.2

Subtract $10 \sqrt{6}$ from both sides of the equation.

$\sqrt{6} x - \sqrt{2} x - 10 \sqrt{2} = 11 - 5 - 10 \sqrt{6}$

Step 3.3

Add $10 \sqrt{2}$ to both sides of the equation.

$\sqrt{6} x - \sqrt{2} x = 11 - 5 - 10 \sqrt{6} + 10 \sqrt{2}$

Step 3.4

Subtract $5$ from $11$ .

$\sqrt{6} x - \sqrt{2} x = 6 - 10 \sqrt{6} + 10 \sqrt{2}$

Step 4

Factor $x$ out of $\sqrt{6} x - \sqrt{2} x$ .

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Step 4.1

Factor $x$ out of $\sqrt{6} x$ .

$x \sqrt{6} - \sqrt{2} x = 6 - 10 \sqrt{6} + 10 \sqrt{2}$

Step 4.2

Factor $x$ out of $- \sqrt{2} x$ .

$x \sqrt{6} + x (- \sqrt{2}) = 6 - 10 \sqrt{6} + 10 \sqrt{2}$

Step 4.3

Factor $x$ out of $x \sqrt{6} + x (- \sqrt{2})$ .

$x (\sqrt{6} - \sqrt{2}) = 6 - 10 \sqrt{6} + 10 \sqrt{2}$

Step 5

Divide each term in $x (\sqrt{6} - \sqrt{2}) = 6 - 10 \sqrt{6} + 10 \sqrt{2}$ by $\sqrt{6} - \sqrt{2}$ and simplify.

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Step 5.1

Divide each term in $x (\sqrt{6} - \sqrt{2}) = 6 - 10 \sqrt{6} + 10 \sqrt{2}$ by $\sqrt{6} - \sqrt{2}$ .

$\frac{x (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{6}{\sqrt{6} - \sqrt{2}} + \frac{- 10 \sqrt{6}}{\sqrt{6} - \sqrt{2}} + \frac{10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\sqrt{6} - \sqrt{2}$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x (\sqrt{6} - \sqrt{2})}{\sqrt{6} - \sqrt{2}} = \frac{6}{\sqrt{6} - \sqrt{2}} + \frac{- 10 \sqrt{6}}{\sqrt{6} - \sqrt{2}} + \frac{10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{\sqrt{6} - \sqrt{2}} + \frac{- 10 \sqrt{6}}{\sqrt{6} - \sqrt{2}} + \frac{10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$x = \frac{6 - 10 \sqrt{6} + 10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$

Step 5.3.2

Multiply $\frac{6 - 10 \sqrt{6} + 10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$x = \frac{6 - 10 \sqrt{6} + 10 \sqrt{2}}{\sqrt{6} - \sqrt{2}} \cdot \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$

Step 5.3.3

Multiply $\frac{6 - 10 \sqrt{6} + 10 \sqrt{2}}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

$x = \frac{(6 - 10 \sqrt{6} + 10 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}$

Step 5.3.4

Expand the denominator using the FOIL method.

$x = \frac{(6 - 10 \sqrt{6} + 10 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{{\sqrt{6}}^{2} + \sqrt{12} - \sqrt{12} - {\sqrt{2}}^{2}}$

Step 5.3.5

Simplify.

$x = \frac{(6 - 10 \sqrt{6} + 10 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{4}$

Step 5.3.6

Cancel the common factor of $6 - 10 \sqrt{6} + 10 \sqrt{2}$ and $4$ .

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Step 5.3.6.1

Factor $2$ out of $(6 - 10 \sqrt{6} + 10 \sqrt{2}) (\sqrt{6} + \sqrt{2})$ .

$x = \frac{2 ((3 - 5 \sqrt{6} + 5 \sqrt{2}) (\sqrt{6} + \sqrt{2}))}{4}$

Step 5.3.6.2

Cancel the common factors.

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Step 5.3.6.2.1

Factor $2$ out of $4$ .

$x = \frac{2 ((3 - 5 \sqrt{6} + 5 \sqrt{2}) (\sqrt{6} + \sqrt{2}))}{2 (2)}$

Step 5.3.6.2.2

Cancel the common factor.

$x = \frac{2 ((3 - 5 \sqrt{6} + 5 \sqrt{2}) (\sqrt{6} + \sqrt{2}))}{2 \cdot 2}$

Step 5.3.6.2.3

Rewrite the expression.

$x = \frac{(3 - 5 \sqrt{6} + 5 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{(3 - 5 \sqrt{6} + 5 \sqrt{2}) (\sqrt{6} + \sqrt{2})}{2}$

Decimal Form:

$x = - 4.20444504 \dots$