Trigonometry Examples

${(- \frac{5}{6})}^{2} + y^{2} = 1^{2}$

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 $- \frac{5}{6}$ .

${(- 1)}^{2} {(\frac{5}{6})}^{2} + y^{2} = 1^{2}$

Step 1.1.2

Apply the product rule to $\frac{5}{6}$ .

${(- 1)}^{2} \frac{5^{2}}{6^{2}} + y^{2} = 1^{2}$

Step 1.2

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

$1 \frac{5^{2}}{6^{2}} + y^{2} = 1^{2}$

Step 1.3

Multiply $\frac{5^{2}}{6^{2}}$ by $1$ .

$\frac{5^{2}}{6^{2}} + y^{2} = 1^{2}$

Step 1.4

Raise $5$ to the power of $2$ .

$\frac{25}{6^{2}} + y^{2} = 1^{2}$

Step 1.5

Raise $6$ to the power of $2$ .

$\frac{25}{36} + y^{2} = 1^{2}$

Step 2

One to any power is one.

$\frac{25}{36} + y^{2} = 1$

Step 3

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

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Step 3.1

Subtract $\frac{25}{36}$ from both sides of the equation.

$y^{2} = 1 - \frac{25}{36}$

Step 3.2

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

$y^{2} = \frac{36}{36} - \frac{25}{36}$

Step 3.3

Combine the numerators over the common denominator.

$y^{2} = \frac{36 - 25}{36}$

Step 3.4

Subtract $25$ from $36$ .

$y^{2} = \frac{11}{36}$

Step 4

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

$y = \pm \sqrt{\frac{11}{36}}$

Step 5

Simplify $\pm \sqrt{\frac{11}{36}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{11}{36}}$ as $\frac{\sqrt{11}}{\sqrt{36}}$ .

$y = \pm \frac{\sqrt{11}}{\sqrt{36}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

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

$y = \pm \frac{\sqrt{11}}{\sqrt{6^{2}}}$

Step 5.2.2

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

$y = \pm \frac{\sqrt{11}}{6}$

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.

$y = \frac{\sqrt{11}}{6}$

Step 6.2

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

$y = - \frac{\sqrt{11}}{6}$

Step 6.3

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

$y = \frac{\sqrt{11}}{6}, - \frac{\sqrt{11}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\sqrt{11}}{6}, - \frac{\sqrt{11}}{6}$

Decimal Form:

$y = 0.55277079 \dots, - 0.55277079 \dots$