Trigonometry Examples

$\frac{a + b}{2} = \sqrt{\frac{a^{2} + b^{2}}{2}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{\frac{a^{2} + b^{2}}{2}} = \frac{a + b}{2}$

Step 2

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

${\sqrt{\frac{a^{2} + b^{2}}{2}}}^{2} = {(\frac{a + b}{2})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}})}^{2} = {(\frac{a + b}{2})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2} \cdot 2} = {(\frac{a + b}{2})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2} \cdot 2} = {(\frac{a + b}{2})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(\frac{a^{2} + b^{2}}{2})}^{1} = {(\frac{a + b}{2})}^{2}$

Step 3.2.1.2

Simplify.

$\frac{a^{2} + b^{2}}{2} = {(\frac{a + b}{2})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\frac{a + b}{2})}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $\frac{a + b}{2}$ .

$\frac{a^{2} + b^{2}}{2} = \frac{{(a + b)}^{2}}{2^{2}}$

Step 3.3.1.2

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

$\frac{a^{2} + b^{2}}{2} = \frac{{(a + b)}^{2}}{4}$

Step 4

Solve for $a$ .

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Step 4.1

Multiply both sides by $2$ .

$\frac{a^{2} + b^{2}}{2} \cdot 2 = \frac{{(a + b)}^{2}}{4} \cdot 2$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{a^{2} + b^{2}}{2} \cdot 2 = \frac{{(a + b)}^{2}}{4} \cdot 2$

Step 4.2.1.1.2

Rewrite the expression.

$a^{2} + b^{2} = \frac{{(a + b)}^{2}}{4} \cdot 2$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Factor $2$ out of $4$ .

$a^{2} + b^{2} = \frac{{(a + b)}^{2}}{2 (2)} \cdot 2$

Step 4.2.2.1.2

Cancel the common factor.

$a^{2} + b^{2} = \frac{{(a + b)}^{2}}{2 \cdot 2} \cdot 2$

Step 4.2.2.1.3

Rewrite the expression.

$a^{2} + b^{2} = \frac{{(a + b)}^{2}}{2}$

Step 4.3

Solve for $a$ .

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Step 4.3.1

Multiply both sides by $2$ .

$(a^{2} + b^{2}) \cdot 2 = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the left side.

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Step 4.3.2.1.1

Simplify $(a^{2} + b^{2}) \cdot 2$ .

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Step 4.3.2.1.1.1

Simplify by multiplying through.

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Step 4.3.2.1.1.1.1

Apply the distributive property.

$a^{2} \cdot 2 + b^{2} \cdot 2 = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2.1.1.1.2

Reorder.

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Step 4.3.2.1.1.1.2.1

Move $2$ to the left of $a^{2}$ .

$2 \cdot a^{2} + b^{2} \cdot 2 = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2.1.1.1.2.2

Move $2$ to the left of $b^{2}$ .

$2 \cdot a^{2} + 2 \cdot b^{2} = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2.1.1.2

Multiply $2$ by $b^{2}$ .

$2 a^{2} + 2 b^{2} = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2.2

Simplify the right side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$2 a^{2} + 2 b^{2} = \frac{{(a + b)}^{2}}{2} \cdot 2$

Step 4.3.2.2.1.2

Rewrite the expression.

$2 a^{2} + 2 b^{2} = {(a + b)}^{2}$

Step 4.3.3

Solve for $a$ .

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Step 4.3.3.1

Simplify ${(a + b)}^{2}$ .

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Step 4.3.3.1.1

Rewrite.

$2 a^{2} + 2 b^{2} = 0 + 0 + {(a + b)}^{2}$

Step 4.3.3.1.2

Rewrite ${(a + b)}^{2}$ as $(a + b) (a + b)$ .

$2 a^{2} + 2 b^{2} = (a + b) (a + b)$

Step 4.3.3.1.3

Expand $(a + b) (a + b)$ using the FOIL Method.

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Step 4.3.3.1.3.1

Apply the distributive property.

$2 a^{2} + 2 b^{2} = a (a + b) + b (a + b)$

Step 4.3.3.1.3.2

Apply the distributive property.

$2 a^{2} + 2 b^{2} = a \cdot a + a b + b (a + b)$

Step 4.3.3.1.3.3

Apply the distributive property.

$2 a^{2} + 2 b^{2} = a \cdot a + a b + b a + b \cdot b$

Step 4.3.3.1.4

Simplify and combine like terms.

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Step 4.3.3.1.4.1

Simplify each term.

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Step 4.3.3.1.4.1.1

Multiply $a$ by $a$ .

$2 a^{2} + 2 b^{2} = a^{2} + a b + b a + b \cdot b$

Step 4.3.3.1.4.1.2

Multiply $b$ by $b$ .

$2 a^{2} + 2 b^{2} = a^{2} + a b + b a + b^{2}$

Step 4.3.3.1.4.2

Add $a b$ and $b a$ .

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Step 4.3.3.1.4.2.1

Reorder $b$ and $a$ .

$2 a^{2} + 2 b^{2} = a^{2} + a b + a b + b^{2}$

Step 4.3.3.1.4.2.2

Add $a b$ and $a b$ .

$2 a^{2} + 2 b^{2} = a^{2} + 2 a b + b^{2}$

Step 4.3.3.2

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$a^{2} + 2 a b + b^{2} = 2 a^{2} + 2 b^{2}$

Step 4.3.3.3

Move all terms containing $a$ to the left side of the equation.

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Step 4.3.3.3.1

Subtract $2 a^{2}$ from both sides of the equation.

$a^{2} + 2 a b + b^{2} - 2 a^{2} = 2 b^{2}$

Step 4.3.3.3.2

Subtract $2 a^{2}$ from $a^{2}$ .

$- a^{2} + 2 a b + b^{2} = 2 b^{2}$

Step 4.3.3.4

Move all terms to the left side of the equation and simplify.

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Step 4.3.3.4.1

Subtract $2 b^{2}$ from both sides of the equation.

$- a^{2} + 2 a b + b^{2} - 2 b^{2} = 0$

Step 4.3.3.4.2

Subtract $2 b^{2}$ from $b^{2}$ .

$- a^{2} + 2 a b - b^{2} = 0$

Step 4.3.3.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3.3.6

Substitute the values $a = - 1$ , $b = 2 b$ , and $c = - b^{2}$ into the quadratic formula and solve for $a$ .

$\frac{- 2 b \pm \sqrt{{(2 b)}^{2} - 4 \cdot (- 1 \cdot (- b^{2}))}}{2 \cdot - 1}$

Step 4.3.3.7

Simplify.

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Step 4.3.3.7.1

Simplify the numerator.

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Step 4.3.3.7.1.1

Rewrite $- 4 \cdot - 1 b^{2}$ as ${(2 b)}^{2}$ .

$a = \frac{- 2 b \pm \sqrt{{(2 b)}^{2} - {(2 b)}^{2}}}{2 \cdot - 1}$

Step 4.3.3.7.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 b$ and $b = 2 b$ .

$a = \frac{- 2 b \pm \sqrt{(2 b + 2 b) (2 b - (2 b))}}{2 \cdot - 1}$

Step 4.3.3.7.1.3

Simplify.

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Step 4.3.3.7.1.3.1

Add $2 b$ and $2 b$ .

$a = \frac{- 2 b \pm \sqrt{4 b (2 b - (2 b))}}{2 \cdot - 1}$

Step 4.3.3.7.1.3.2

Multiply $2$ by $- 1$ .

$a = \frac{- 2 b \pm \sqrt{4 b (2 b - 2 b)}}{2 \cdot - 1}$

Step 4.3.3.7.1.4

Subtract $2 b$ from $2 b$ .

$a = \frac{- 2 b \pm \sqrt{4 b \cdot 0}}{2 \cdot - 1}$

Step 4.3.3.7.1.5

Combine exponents.

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Step 4.3.3.7.1.5.1

Multiply $0$ by $4$ .

$a = \frac{- 2 b \pm \sqrt{0 b}}{2 \cdot - 1}$

Step 4.3.3.7.1.5.2

Multiply $0$ by $b$ .

$a = \frac{- 2 b \pm \sqrt{0}}{2 \cdot - 1}$

Step 4.3.3.7.1.6

Rewrite $0$ as $0^{2}$ .

$a = \frac{- 2 b \pm \sqrt{0^{2}}}{2 \cdot - 1}$

Step 4.3.3.7.1.7

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

$a = \frac{- 2 b \pm 0}{2 \cdot - 1}$

Step 4.3.3.7.1.8

$- 2 b$ plus or minus $0$ is $- 2 b$ .

$a = \frac{- 2 b}{2 \cdot - 1}$

Step 4.3.3.7.2

Multiply $2$ by $- 1$ .

$a = \frac{- 2 b}{- 2}$

Step 4.3.3.7.3

Cancel the common factor of $- 2$ .

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Step 4.3.3.7.3.1

Cancel the common factor.

$a = \frac{- 2 b}{- 2}$

Step 4.3.3.7.3.2

Divide $b$ by $1$ .

$a = b$

Step 4.3.3.8

The final answer is the combination of both solutions.

$a = b$ Double roots