Trigonometry Examples

$\cos (22.5) = (\frac{1}{2} \cdot \sqrt{A + \sqrt{B}})$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot \sqrt{A + \sqrt{B}} = \cos (22.5)$ .

$\frac{1}{2} \cdot \sqrt{A + \sqrt{B}} = \cos (22.5)$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot \sqrt{A + \sqrt{B}}) = 2 \cos (22.5)$

Step 3

Simplify both sides of the equation.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Simplify $2 (\frac{1}{2} \cdot \sqrt{A + \sqrt{B}})$ .

Tap for more steps...

Step 3.1.1.1

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

$2 \frac{\sqrt{A + \sqrt{B}}}{2} = 2 \cos (22.5)$

Step 3.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.1.1.2.1

Cancel the common factor.

$2 \frac{\sqrt{A + \sqrt{B}}}{2} = 2 \cos (22.5)$

Step 3.1.1.2.2

Rewrite the expression.

$\sqrt{A + \sqrt{B}} = 2 \cos (22.5)$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $2 \cos (22.5)$ .

Tap for more steps...

Step 3.2.1.1

The exact value of $\cos (22.5)$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

Tap for more steps...

Step 3.2.1.1.1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sqrt{A + \sqrt{B}} = 2 \cos (\frac{45}{2})$

Step 3.2.1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\sqrt{A + \sqrt{B}} = 2 (\pm \sqrt{\frac{1 + \cos (45)}{2}})$

Step 3.2.1.1.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$\sqrt{A + \sqrt{B}} = 2 \sqrt{\frac{1 + \cos (45)}{2}}$

Step 3.2.1.1.4

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

$\sqrt{A + \sqrt{B}} = 2 \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

Step 3.2.1.1.5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

Tap for more steps...

Step 3.2.1.1.5.1

Write $1$ as a fraction with a common denominator.

$\sqrt{A + \sqrt{B}} = 2 \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}$

Step 3.2.1.1.5.2

Combine the numerators over the common denominator.

$\sqrt{A + \sqrt{B}} = 2 \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}$

Step 3.2.1.1.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2.1.1.5.4

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

Tap for more steps...

Step 3.2.1.1.5.4.1

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

$\sqrt{A + \sqrt{B}} = 2 \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}}$

Step 3.2.1.1.5.4.2

Multiply $2$ by $2$ .

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

Step 3.2.1.1.5.5

Rewrite $\sqrt{\frac{2 + \sqrt{2}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}}$ .

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

Step 3.2.1.1.5.6

Simplify the denominator.

Tap for more steps...

Step 3.2.1.1.5.6.1

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

$\sqrt{A + \sqrt{B}} = 2 \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}}$

Step 3.2.1.1.5.6.2

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

$\sqrt{A + \sqrt{B}} = 2 \frac{\sqrt{2 + \sqrt{2}}}{2}$

Step 3.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.2.1

Cancel the common factor.

$\sqrt{A + \sqrt{B}} = 2 \frac{\sqrt{2 + \sqrt{2}}}{2}$

Step 3.2.1.2.2

Rewrite the expression.

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

Step 4

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

${\sqrt{A + \sqrt{B}}}^{2} = {\sqrt{2 + \sqrt{2}}}^{2}$

Step 5

Simplify each side of the equation.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

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

Tap for more steps...

Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

${(A + \sqrt{B})}^{1} = {\sqrt{2 + \sqrt{2}}}^{2}$

Step 5.2.1.2

Simplify.

$A + \sqrt{B} = {\sqrt{2 + \sqrt{2}}}^{2}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

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

Tap for more steps...

Step 5.3.1.1

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

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

Step 5.3.1.2

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

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

Step 5.3.1.3

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

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

Step 5.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.1.4.1

Cancel the common factor.

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

Step 5.3.1.4.2

Rewrite the expression.

$A + \sqrt{B} = {(2 + \sqrt{2})}^{1}$

Step 5.3.1.5

Simplify.

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

Step 6

Subtract $A$ from both sides of the equation.

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

Step 7

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

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

Step 8

Simplify each side of the equation.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

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

Tap for more steps...

Step 8.2.1.1

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

Tap for more steps...

Step 8.2.1.1.1

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

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

Step 8.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.1.2.1

Cancel the common factor.

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

Step 8.2.1.1.2.2

Rewrite the expression.

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

Step 8.2.1.2

Simplify.

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

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify ${(2 + \sqrt{2} - A)}^{2}$ .

Tap for more steps...

Step 8.3.1.1

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

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

Step 8.3.1.2

Expand $(2 + \sqrt{2} - A) (2 + \sqrt{2} - A)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 8.3.1.3

Simplify terms.

Tap for more steps...

Step 8.3.1.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.3.1.1

Multiply $2$ by $2$ .

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

Step 8.3.1.3.1.2

Multiply $- 1$ by $2$ .

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

Step 8.3.1.3.1.3

Move $2$ to the left of $\sqrt{2}$ .

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

Step 8.3.1.3.1.4

Combine using the product rule for radicals.

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

Step 8.3.1.3.1.5

Multiply $2$ by $2$ .

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

Step 8.3.1.3.1.6

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

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

Step 8.3.1.3.1.7

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

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

Step 8.3.1.3.1.8

Multiply $2$ by $- 1$ .

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

Step 8.3.1.3.1.9

Rewrite using the commutative property of multiplication.

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

Step 8.3.1.3.1.10

Multiply $A$ by $A$ by adding the exponents.

Tap for more steps...

Step 8.3.1.3.1.10.1

Move $A$ .

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

Step 8.3.1.3.1.10.2

Multiply $A$ by $A$ .

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

Step 8.3.1.3.1.11

Multiply $- 1$ by $- 1$ .

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

Step 8.3.1.3.1.12

Multiply $A^{2}$ by $1$ .

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

Step 8.3.1.3.2

Simplify by adding terms.

Tap for more steps...

Step 8.3.1.3.2.1

Add $4$ and $2$ .

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

Step 8.3.1.3.2.2

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

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

Step 8.3.1.3.2.3

Subtract $2 A$ from $- 2 A$ .

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

Step 8.3.1.3.2.4

Reorder the factors of $\sqrt{2} (- A)$ .

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

Step 8.3.1.3.2.5

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

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