Precalculus Examples

$\sin (15) = (\frac{1}{2}) \sqrt{A - \sqrt{B}}$

Step 1

Simplify the left side.

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Step 1.1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 - 30) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.2

Separate negation.

$\sin (45 - (30)) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.3

Apply the difference of angles identity.

$\sin (45) \cos (30) - \cos (45) \sin (30) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30) = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8

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

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

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

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Step 1.1.8.1.1.1

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

$\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

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

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 1.1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{1}{2} \cdot \sqrt{A - \sqrt{B}}$

Step 2

Simplify the right side.

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Step 2.1

Combine $\frac{1}{2}$ and $\sqrt{A - \sqrt{B}}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{A - \sqrt{B}}}{2}$

Step 3

Rewrite the equation as $\frac{\sqrt{A - \sqrt{B}}}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$ .

$\frac{\sqrt{A - \sqrt{B}}}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 4

Multiply both sides by $2$ .

$\frac{\sqrt{A - \sqrt{B}}}{2} \cdot 2 = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot 2$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{\sqrt{A - \sqrt{B}}}{2} \cdot 2 = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$\sqrt{A - \sqrt{B}} = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Factor $2$ out of $4$ .

$\sqrt{A - \sqrt{B}} = \frac{\sqrt{6} - \sqrt{2}}{2 (2)} \cdot 2$

Step 5.2.1.2

Cancel the common factor.

$\sqrt{A - \sqrt{B}} = \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2} \cdot 2$

Step 5.2.1.3

Rewrite the expression.

$\sqrt{A - \sqrt{B}} = \frac{\sqrt{6} - \sqrt{2}}{2}$

Step 6

Solve for $A$ .

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Step 6.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{A - \sqrt{B}}}^{2} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

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

${({(A - \sqrt{B})}^{\frac{1}{2}})}^{2} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify ${({(A - \sqrt{B})}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1

Multiply the exponents in ${({(A - \sqrt{B})}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

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

${(A - \sqrt{B})}^{\frac{1}{2} \cdot 2} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(A - \sqrt{B})}^{\frac{1}{2} \cdot 2} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(A - \sqrt{B})}^{1} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2.2.1.2

Simplify.

$A - \sqrt{B} = {(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify ${(\frac{\sqrt{6} - \sqrt{2}}{2})}^{2}$ .

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Step 6.2.3.1.1

Simplify the expression.

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Step 6.2.3.1.1.1

Apply the product rule to $\frac{\sqrt{6} - \sqrt{2}}{2}$ .

$A - \sqrt{B} = \frac{{(\sqrt{6} - \sqrt{2})}^{2}}{2^{2}}$

Step 6.2.3.1.1.2

Raise $2$ to the power of $2$ .

$A - \sqrt{B} = \frac{{(\sqrt{6} - \sqrt{2})}^{2}}{4}$

Step 6.2.3.1.1.3

Rewrite ${(\sqrt{6} - \sqrt{2})}^{2}$ as $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ .

$A - \sqrt{B} = \frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})}{4}$

Step 6.2.3.1.2

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

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Step 6.2.3.1.2.1

Apply the distributive property.

$A - \sqrt{B} = \frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{4}$

Step 6.2.3.1.2.2

Apply the distributive property.

$A - \sqrt{B} = \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{4}$

Step 6.2.3.1.2.3

Apply the distributive property.

$A - \sqrt{B} = \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3

Simplify and combine like terms.

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Step 6.2.3.1.3.1

Simplify each term.

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Step 6.2.3.1.3.1.1

Combine using the product rule for radicals.

$A - \sqrt{B} = \frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.2

Multiply $6$ by $6$ .

$A - \sqrt{B} = \frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.3

Rewrite $36$ as $6^{2}$ .

$A - \sqrt{B} = \frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.4

Pull terms out from under the radical, assuming positive real numbers.

$A - \sqrt{B} = \frac{6 + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.5

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 6.2.3.1.3.1.5.1

Combine using the product rule for radicals.

$A - \sqrt{B} = \frac{6 - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.5.2

Multiply $6$ by $2$ .

$A - \sqrt{B} = \frac{6 - \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.6

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.2.3.1.3.1.6.1

Factor $4$ out of $12$ .

$A - \sqrt{B} = \frac{6 - \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.6.2

Rewrite $4$ as $2^{2}$ .

$A - \sqrt{B} = \frac{6 - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.7

Pull terms out from under the radical.

$A - \sqrt{B} = \frac{6 - (2 \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.8

Multiply $2$ by $- 1$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.9

Multiply $- \sqrt{2} \sqrt{6}$ .

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Step 6.2.3.1.3.1.9.1

Combine using the product rule for radicals.

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.9.2

Multiply $6$ by $2$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - \sqrt{12} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.10

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.2.3.1.3.1.10.1

Factor $4$ out of $12$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.10.2

Rewrite $4$ as $2^{2}$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.11

Pull terms out from under the radical.

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.12

Multiply $2$ by $- 1$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} (- \sqrt{2})}{4}$

Step 6.2.3.1.3.1.13

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 6.2.3.1.3.1.13.1

Multiply $- 1$ by $- 1$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2}}{4}$

Step 6.2.3.1.3.1.13.2

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{4}$

Step 6.2.3.1.3.1.13.3

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2}}{4}$

Step 6.2.3.1.3.1.13.4

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{4}$

Step 6.2.3.1.3.1.13.5

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1}}{4}$

Step 6.2.3.1.3.1.13.6

Add $1$ and $1$ .

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{2}}{4}$

Step 6.2.3.1.3.1.14

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

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Step 6.2.3.1.3.1.14.1

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2}}{4}$

Step 6.2.3.1.3.1.14.2

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2}}{4}$

Step 6.2.3.1.3.1.14.3

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

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{4}$

Step 6.2.3.1.3.1.14.4

Cancel the common factor of $2$ .

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Step 6.2.3.1.3.1.14.4.1

Cancel the common factor.

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{4}$

Step 6.2.3.1.3.1.14.4.2

Rewrite the expression.

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1}}{4}$

Step 6.2.3.1.3.1.14.5

Evaluate the exponent.

$A - \sqrt{B} = \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2}{4}$

Step 6.2.3.1.3.2

Add $6$ and $2$ .

$A - \sqrt{B} = \frac{8 - 2 \sqrt{3} - 2 \sqrt{3}}{4}$

Step 6.2.3.1.3.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$A - \sqrt{B} = \frac{8 - 4 \sqrt{3}}{4}$

Step 6.2.3.1.4

Cancel the common factor of $8 - 4 \sqrt{3}$ and $4$ .

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Step 6.2.3.1.4.1

Factor $4$ out of $8$ .

$A - \sqrt{B} = \frac{4 \cdot 2 - 4 \sqrt{3}}{4}$

Step 6.2.3.1.4.2

Factor $4$ out of $- 4 \sqrt{3}$ .

$A - \sqrt{B} = \frac{4 \cdot 2 + 4 (- \sqrt{3})}{4}$

Step 6.2.3.1.4.3

Factor $4$ out of $4 (2) + 4 (- \sqrt{3})$ .

$A - \sqrt{B} = \frac{4 (2 - \sqrt{3})}{4}$

Step 6.2.3.1.4.4

Cancel the common factors.

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Step 6.2.3.1.4.4.1

Factor $4$ out of $4$ .

$A - \sqrt{B} = \frac{4 (2 - \sqrt{3})}{4 (1)}$

Step 6.2.3.1.4.4.2

Cancel the common factor.

$A - \sqrt{B} = \frac{4 (2 - \sqrt{3})}{4 \cdot 1}$

Step 6.2.3.1.4.4.3

Rewrite the expression.

$A - \sqrt{B} = \frac{2 - \sqrt{3}}{1}$

Step 6.2.3.1.4.4.4

Divide $2 - \sqrt{3}$ by $1$ .

$A - \sqrt{B} = 2 - \sqrt{3}$

Step 6.3

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

$A = 2 - \sqrt{3} + \sqrt{B}$