Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2

Raise $- 1$ to the power of $2$ .

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

Step 1.3

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

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

Step 1.4

One to any power is one.

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

Step 1.5

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

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

Step 2

One to any power is one.

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

Step 3

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

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Step 3.1

Subtract $\frac{1}{4}$ from both sides of the equation.

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Subtract $1$ from $4$ .

$y^{2} = \frac{3}{4}$

Step 4

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

$y = \pm \sqrt{\frac{3}{4}}$

Step 5

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$y = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

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

$y = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 5.2.2

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

$y = \pm \frac{\sqrt{3}}{2}$

Step 6

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

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Step 6.1

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

$y = \frac{\sqrt{3}}{2}$

Step 6.2

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

$y = - \frac{\sqrt{3}}{2}$

Step 6.3

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

$y = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}$

Decimal Form:

$y = 0.86602540 \dots, - 0.86602540 \dots$