Trigonometry Examples

$(\frac{- \sqrt{2}}{5}, \frac{\sqrt{2}}{2})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{- \sqrt{2}}{5}, \frac{\sqrt{2}}{2})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{- \sqrt{2}}{5}, 0)$ , and $(\frac{- \sqrt{2}}{5}, \frac{\sqrt{2}}{2})$ .

Opposite : $\frac{\sqrt{2}}{2}$

Adjacent : $\frac{- \sqrt{2}}{5}$

Step 2

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

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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Step 2.2.1

Apply the product rule to $- \frac{\sqrt{2}}{5}$ .

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

Step 2.2.2

Apply the product rule to $\frac{\sqrt{2}}{5}$ .

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

Step 2.3

Simplify the expression.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Evaluate the exponent.

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

Step 2.5

Simplify the expression.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

$\sqrt{\frac{2}{25} + \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{2}{25} + \frac{2^{1}}{2^{2}}}$

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

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

$\sqrt{\frac{2}{25} + \frac{2}{4}}$

Step 2.8

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

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Step 2.8.1

Factor $2$ out of $2$ .

$\sqrt{\frac{2}{25} + \frac{2 (1)}{4}}$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $4$ .

$\sqrt{\frac{2}{25} + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.8.2.2

Cancel the common factor.

$\sqrt{\frac{2}{25} + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.8.2.3

Rewrite the expression.

$\sqrt{\frac{2}{25} + \frac{1}{2}}$

Step 2.9

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

$\sqrt{\frac{2}{25} \cdot \frac{2}{2} + \frac{1}{2}}$

Step 2.10

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\sqrt{\frac{2}{25} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{25}{25}}$

Step 2.11

Write each expression with a common denominator of $50$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

Multiply $\frac{2}{25}$ by $\frac{2}{2}$ .

$\sqrt{\frac{2 \cdot 2}{25 \cdot 2} + \frac{1}{2} \cdot \frac{25}{25}}$

Step 2.11.2

Multiply $25$ by $2$ .

$\sqrt{\frac{2 \cdot 2}{50} + \frac{1}{2} \cdot \frac{25}{25}}$

Step 2.11.3

Multiply $\frac{1}{2}$ by $\frac{25}{25}$ .

$\sqrt{\frac{2 \cdot 2}{50} + \frac{25}{2 \cdot 25}}$

Step 2.11.4

Multiply $2$ by $25$ .

$\sqrt{\frac{2 \cdot 2}{50} + \frac{25}{50}}$

Step 2.12

Combine the numerators over the common denominator.

$\sqrt{\frac{2 \cdot 2 + 25}{50}}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Multiply $2$ by $2$ .

$\sqrt{\frac{4 + 25}{50}}$

Step 2.13.2

Add $4$ and $25$ .

$\sqrt{\frac{29}{50}}$

Step 2.14

Rewrite $\sqrt{\frac{29}{50}}$ as $\frac{\sqrt{29}}{\sqrt{50}}$ .

$\frac{\sqrt{29}}{\sqrt{50}}$

Step 2.15

Simplify the denominator.

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Step 2.15.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.15.1.1

Factor $25$ out of $50$ .

$\frac{\sqrt{29}}{\sqrt{25 (2)}}$

Step 2.15.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{29}}{\sqrt{5^{2} \cdot 2}}$

Step 2.15.2

Pull terms out from under the radical.

$\frac{\sqrt{29}}{5 \sqrt{2}}$

Step 2.16

Multiply $\frac{\sqrt{29}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{29}}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.17

Combine and simplify the denominator.

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Step 2.17.1

Multiply $\frac{\sqrt{29}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{29} \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Step 2.17.2

Move $\sqrt{2}$ .

$\frac{\sqrt{29} \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 2.17.3

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

$\frac{\sqrt{29} \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 2.17.4

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

$\frac{\sqrt{29} \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 2.17.5

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

$\frac{\sqrt{29} \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Step 2.17.6

Add $1$ and $1$ .

$\frac{\sqrt{29} \sqrt{2}}{5 {\sqrt{2}}^{2}}$

Step 2.17.7

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

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Step 2.17.7.1

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

$\frac{\sqrt{29} \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

Step 2.17.7.2

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

$\frac{\sqrt{29} \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.17.7.3

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

$\frac{\sqrt{29} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 2.17.7.4

Cancel the common factor of $2$ .

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Step 2.17.7.4.1

Cancel the common factor.

$\frac{\sqrt{29} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 2.17.7.4.2

Rewrite the expression.

$\frac{\sqrt{29} \sqrt{2}}{5 \cdot 2^{1}}$

Step 2.17.7.5

Evaluate the exponent.

$\frac{\sqrt{29} \sqrt{2}}{5 \cdot 2}$

Step 2.18

Simplify the numerator.

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Step 2.18.1

Combine using the product rule for radicals.

$\frac{\sqrt{29 \cdot 2}}{5 \cdot 2}$

Step 2.18.2

Multiply $29$ by $2$ .

$\frac{\sqrt{58}}{5 \cdot 2}$

Step 2.19

Multiply $5$ by $2$ .

$\frac{\sqrt{58}}{10}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{- \sqrt{2}}{5}}{\frac{\sqrt{58}}{10}}$ .

$\frac{\frac{- \sqrt{2}}{5}}{\frac{\sqrt{58}}{10}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{- \sqrt{2}}{5} \cdot \frac{10}{\sqrt{58}}$

Step 4.2

Cancel the common factor of $5$ .

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Step 4.2.1

Factor $5$ out of $10$ .

$\cos (θ) = \frac{- \sqrt{2}}{5} \cdot \frac{5 (2)}{\sqrt{58}}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{- \sqrt{2}}{5} \cdot \frac{5 \cdot 2}{\sqrt{58}}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = - \sqrt{2} \frac{2}{\sqrt{58}}$

Step 4.3

Combine $\frac{2}{\sqrt{58}}$ and $\sqrt{2}$ .

$\cos (θ) = - \frac{2 \sqrt{2}}{\sqrt{58}}$

Step 4.4

Combine $\sqrt{2}$ and $\sqrt{58}$ into a single radical.

$\cos (θ) = - (2 \sqrt{\frac{2}{58}})$

Step 4.5

Cancel the common factor of $2$ and $58$ .

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Step 4.5.1

Factor $2$ out of $2$ .

$\cos (θ) = - (2 \sqrt{\frac{2 (1)}{58}})$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $2$ out of $58$ .

$\cos (θ) = - (2 \sqrt{\frac{2 \cdot 1}{2 \cdot 29}})$

Step 4.5.2.2

Cancel the common factor.

$\cos (θ) = - (2 \sqrt{\frac{2 \cdot 1}{2 \cdot 29}})$

Step 4.5.2.3

Rewrite the expression.

$\cos (θ) = - (2 \sqrt{\frac{1}{29}})$

Step 4.6

Rewrite $\sqrt{\frac{1}{29}}$ as $\frac{\sqrt{1}}{\sqrt{29}}$ .

$\cos (θ) = - (2 (\frac{\sqrt{1}}{\sqrt{29}}))$

Step 4.7

Any root of $1$ is $1$ .

$\cos (θ) = - (2 (\frac{1}{\sqrt{29}}))$

Step 4.8

Multiply $\frac{1}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\cos (θ) = - (2 (\frac{1}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}}))$

Step 4.9

Combine and simplify the denominator.

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Step 4.9.1

Multiply $\frac{1}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\cos (θ) = - (2 (\frac{\sqrt{29}}{\sqrt{29} \sqrt{29}}))$

Step 4.9.2

Raise $\sqrt{29}$ to the power of $1$ .

$\cos (θ) = - (2 (\frac{\sqrt{29}}{\sqrt{29} \sqrt{29}}))$

Step 4.9.3

Raise $\sqrt{29}$ to the power of $1$ .

$\cos (θ) = - (2 (\frac{\sqrt{29}}{\sqrt{29} \sqrt{29}}))$

Step 4.9.4

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

$\cos (θ) = - (2 (\frac{\sqrt{29}}{{\sqrt{29}}^{1 + 1}}))$

Step 4.9.5

Add $1$ and $1$ .

$\cos (θ) = - (2 (\frac{\sqrt{29}}{{\sqrt{29}}^{2}}))$

Step 4.9.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 4.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\cos (θ) = - (2 (\frac{\sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}))$

Step 4.9.6.2

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

$\cos (θ) = - (2 (\frac{\sqrt{29}}{29^{\frac{1}{2} \cdot 2}}))$

Step 4.9.6.3

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

$\cos (θ) = - (2 (\frac{\sqrt{29}}{29^{\frac{2}{2}}}))$

Step 4.9.6.4

Cancel the common factor of $2$ .

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Step 4.9.6.4.1

Cancel the common factor.

$\cos (θ) = - (2 (\frac{\sqrt{29}}{29^{\frac{2}{2}}}))$

Step 4.9.6.4.2

Rewrite the expression.

$\cos (θ) = - (2 (\frac{\sqrt{29}}{29}))$

Step 4.9.6.5

Evaluate the exponent.

$\cos (θ) = - (2 (\frac{\sqrt{29}}{29}))$

Step 4.10

Combine $2$ and $\frac{\sqrt{29}}{29}$ .

$\cos (θ) = - \frac{2 \sqrt{29}}{29}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{2 \sqrt{29}}{29} \approx - 0.37139067$