Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

$\frac{{\sqrt{2}}^{2}}{2^{2}} + y^{2} = 1$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} + y^{2} = 1$

Step 1.2.3

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

$\frac{2^{\frac{2}{2}}}{2^{2}} + y^{2} = 1$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}} + y^{2} = 1$

Step 1.2.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}} + y^{2} = 1$

Step 1.2.5

Evaluate the exponent.

$\frac{2}{2^{2}} + y^{2} = 1$

Step 1.3

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

$\frac{2}{4} + y^{2} = 1$

Step 1.4

Cancel the common factor of $2$ and $4$ .

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Step 1.4.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} + y^{2} = 1$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} + y^{2} = 1$

Step 1.4.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} + y^{2} = 1$

Step 1.4.2.3

Rewrite the expression.

$\frac{1}{2} + y^{2} = 1$

Step 2

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

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Step 2.1

Subtract $\frac{1}{2}$ from both sides of the equation.

$y^{2} = 1 - \frac{1}{2}$

Step 2.2

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

$y^{2} = \frac{2}{2} - \frac{1}{2}$

Step 2.3

Combine the numerators over the common denominator.

$y^{2} = \frac{2 - 1}{2}$

Step 2.4

Subtract $1$ from $2$ .

$y^{2} = \frac{1}{2}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{1}{2}}$

Step 4

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

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Step 4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 4.2

Any root of $1$ is $1$ .

$y = \pm \frac{1}{\sqrt{2}}$

Step 4.3

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

$y = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

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

$y = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.4.2

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

$y = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.4.3

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

$y = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.4.4

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

$y = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.4.6

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

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Step 4.4.6.1

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

$y = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$y = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$y = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2}}{2^{1}}$

Step 4.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2}}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{2}}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{2}}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Decimal Form:

$y = 0.70710678 \dots, - 0.70710678 \dots$