Trigonometry Examples

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

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{\sqrt{5}}, - \frac{2}{\sqrt{5}})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{\sqrt{5}}, 0)$ , and $(\frac{1}{\sqrt{5}}, - \frac{2}{\sqrt{5}})$ .

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

Adjacent : $\frac{1}{\sqrt{5}}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.2.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

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

Step 2.2.6.4.2

Rewrite the expression.

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

Step 2.2.6.5

Evaluate the exponent.

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

Step 2.3

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

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

Step 2.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Evaluate the exponent.

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

Step 2.5

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

$\sqrt{\frac{5}{25} + {(- \frac{2}{\sqrt{5}})}^{2}}$

Step 2.6

Cancel the common factor of $5$ and $25$ .

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Step 2.6.1

Factor $5$ out of $5$ .

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $5$ out of $25$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.7

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

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

Step 2.8

Combine and simplify the denominator.

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Step 2.8.1

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

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

Step 2.8.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.8.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.8.4

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

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

Step 2.8.5

Add $1$ and $1$ .

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

Step 2.8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 2.8.6.2

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

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

Step 2.8.6.3

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

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

Step 2.8.6.4

Cancel the common factor of $2$ .

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Step 2.8.6.4.1

Cancel the common factor.

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

Step 2.8.6.4.2

Rewrite the expression.

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

Step 2.8.6.5

Evaluate the exponent.

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

Step 2.9

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

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Step 2.9.1

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

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

Step 2.9.2

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

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

Step 2.9.3

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

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

Step 2.10

Simplify the expression.

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Step 2.10.1

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

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

Step 2.10.2

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

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

Step 2.11

Simplify the numerator.

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Step 2.11.1

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

$\sqrt{\frac{1}{5} + \frac{4 {\sqrt{5}}^{2}}{5^{2}}}$

Step 2.11.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.11.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{5} + \frac{4 {(5^{\frac{1}{2}})}^{2}}{5^{2}}}$

Step 2.11.2.2

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

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5^{\frac{1}{2} \cdot 2}}{5^{2}}}$

Step 2.11.2.3

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

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5^{\frac{2}{2}}}{5^{2}}}$

Step 2.11.2.4

Cancel the common factor of $2$ .

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Step 2.11.2.4.1

Cancel the common factor.

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5^{\frac{2}{2}}}{5^{2}}}$

Step 2.11.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5^{1}}{5^{2}}}$

Step 2.11.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5}{5^{2}}}$

Step 2.12

Reduce the expression by cancelling the common factors.

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Step 2.12.1

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

$\sqrt{\frac{1}{5} + \frac{4 \cdot 5}{25}}$

Step 2.12.2

Multiply $4$ by $5$ .

$\sqrt{\frac{1}{5} + \frac{20}{25}}$

Step 2.12.3

Cancel the common factor of $20$ and $25$ .

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Step 2.12.3.1

Factor $5$ out of $20$ .

$\sqrt{\frac{1}{5} + \frac{5 (4)}{25}}$

Step 2.12.3.2

Cancel the common factors.

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Step 2.12.3.2.1

Factor $5$ out of $25$ .

$\sqrt{\frac{1}{5} + \frac{5 \cdot 4}{5 \cdot 5}}$

Step 2.12.3.2.2

Cancel the common factor.

$\sqrt{\frac{1}{5} + \frac{5 \cdot 4}{5 \cdot 5}}$

Step 2.12.3.2.3

Rewrite the expression.

$\sqrt{\frac{1}{5} + \frac{4}{5}}$

Step 2.12.4

Simplify the expression.

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Step 2.12.4.1

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 4}{5}}$

Step 2.12.4.2

Add $1$ and $4$ .

$\sqrt{\frac{5}{5}}$

Step 2.12.4.3

Divide $5$ by $5$ .

$\sqrt{1}$

Step 2.12.4.4

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{1}{\sqrt{5}}}{1}$ .

$\frac{\frac{1}{\sqrt{5}}}{1}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Divide $\frac{1}{\sqrt{5}}$ by $1$ .

$\cos (θ) = \frac{1}{\sqrt{5}}$

Step 4.2

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

$\cos (θ) = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

$\cos (θ) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.3.4

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

$\cos (θ) = \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\cos (θ) = \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{\sqrt{5}}{5}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = \frac{\sqrt{5}}{5}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{5}}{5} \approx 0.44721359$