Algebra Examples

$(- \frac{3}{5}, y)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{3}{5}, y)$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{3}{5}, 0)$ , and $(- \frac{3}{5}, y)$ .

Opposite : $y$

Adjacent : $- \frac{3}{5}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

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

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Step 2.1.1

Apply the product rule to $- \frac{3}{5}$ .

$\sqrt{{(- 1)}^{2} {(\frac{3}{5})}^{2} + {(y)}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{3}{5}$ .

$\sqrt{{(- 1)}^{2} \frac{3^{2}}{5^{2}} + {(y)}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{3^{2}}{5^{2}} + {(y)}^{2}}$

Step 2.3

Multiply $\frac{3^{2}}{5^{2}}$ by $1$ .

$\sqrt{\frac{3^{2}}{5^{2}} + {(y)}^{2}}$

Step 2.4

Raise $3$ to the power of $2$ .

$\sqrt{\frac{9}{5^{2}} + {(y)}^{2}}$

Step 2.5

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

$\sqrt{\frac{9}{25} + {(y)}^{2}}$

Step 2.6

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\sqrt{\frac{9}{25} + y^{2} \cdot \frac{25}{25}}$

Step 2.7

Simplify terms.

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Step 2.7.1

Combine $y^{2}$ and $\frac{25}{25}$ .

$\sqrt{\frac{9}{25} + \frac{y^{2} \cdot 25}{25}}$

Step 2.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + y^{2} \cdot 25}{25}}$

Step 2.8

Move $25$ to the left of $y^{2}$ .

$\sqrt{\frac{9 + 25 y^{2}}{25}}$

Step 2.9

Rewrite $\frac{9 + 25 y^{2}}{25}$ as ${(\frac{1}{5})}^{2} (9 + 25 y^{2})$ .

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Step 2.9.1

Factor the perfect power $1^{2}$ out of $9 + 25 y^{2}$ .

$\sqrt{\frac{1^{2} (9 + 25 y^{2})}{25}}$

Step 2.9.2

Factor the perfect power $5^{2}$ out of $25$ .

$\sqrt{\frac{1^{2} (9 + 25 y^{2})}{5^{2} \cdot 1}}$

Step 2.9.3

Rearrange the fraction $\frac{1^{2} (9 + 25 y^{2})}{5^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{5})}^{2} (9 + 25 y^{2})}$

Step 2.10

Pull terms out from under the radical.

$\frac{1}{5} \sqrt{9 + 25 y^{2}}$

Step 2.11

Combine $\frac{1}{5}$ and $\sqrt{9 + 25 y^{2}}$ .

$\frac{\sqrt{9 + 25 y^{2}}}{5}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{3}{5}}{\frac{\sqrt{9 + 25 y^{2}}}{5}}$ .

$\frac{- \frac{3}{5}}{\frac{\sqrt{9 + 25 y^{2}}}{5}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - \frac{3}{5} \cdot \frac{5}{\sqrt{9 + 25 y^{2}}}$

Step 4.2

Cancel the common factor of $5$ .

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Step 4.2.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$\cos (θ) = \frac{- 3}{5} \cdot \frac{5}{\sqrt{9 + 25 y^{2}}}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{- 3}{5} \cdot \frac{5}{\sqrt{9 + 25 y^{2}}}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = - 3 \frac{1}{\sqrt{9 + 25 y^{2}}}$

Step 4.3

Combine $- 3$ and $\frac{1}{\sqrt{9 + 25 y^{2}}}$ .

$\cos (θ) = \frac{- 3}{\sqrt{9 + 25 y^{2}}}$

Step 4.4

Move the negative in front of the fraction.

$\cos (θ) = - \frac{3}{\sqrt{9 + 25 y^{2}}}$

Step 4.5

Multiply $\frac{3}{\sqrt{9 + 25 y^{2}}}$ by $\frac{\sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}}}$ .

$\cos (θ) = - (\frac{3}{\sqrt{9 + 25 y^{2}}} \cdot \frac{\sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{3}{\sqrt{9 + 25 y^{2}}}$ by $\frac{\sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}}}$ .

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}} \sqrt{9 + 25 y^{2}}}$

Step 4.6.2

Raise $\sqrt{9 + 25 y^{2}}$ to the power of $1$ .

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}} \sqrt{9 + 25 y^{2}}}$

Step 4.6.3

Raise $\sqrt{9 + 25 y^{2}}$ to the power of $1$ .

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{\sqrt{9 + 25 y^{2}} \sqrt{9 + 25 y^{2}}}$

Step 4.6.4

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

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{\sqrt{9 + 25 y^{2}}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{\sqrt{9 + 25 y^{2}}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{9 + 25 y^{2}}}^{2}$ as $9 + 25 y^{2}$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 25 y^{2}}$ as ${(9 + 25 y^{2})}^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{({(9 + 25 y^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{(9 + 25 y^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{(9 + 25 y^{2})}^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{{(9 + 25 y^{2})}^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{9 + 25 y^{2}}$

Step 4.6.6.5

Simplify.

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{9 + 25 y^{2}}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{3 \sqrt{9 + 25 y^{2}}}{9 + 25 y^{2}} \approx - \frac{3 \sqrt{9 + 25 y^{2}}}{9 + 25 y^{2}}$