Trigonometry Examples

(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})

$(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})$

Step 1

To find the

cos (θ)

$\cos (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})

$(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(\frac{\sqrt{10}}{10}, 0)

$(\frac{\sqrt{10}}{10}, 0)$ , and

(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})

$(\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10})$ .

Opposite :

\frac{3 \sqrt{10}}{10}

$\frac{3 \sqrt{10}}{10}$

Adjacent :

\frac{\sqrt{10}}{10}

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

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Apply the product rule to

\frac{\sqrt{10}}{10}

$\frac{\sqrt{10}}{10}$ .

\sqrt{\frac{{\sqrt{10}}^{2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}

$\sqrt{\frac{{\sqrt{10}}^{2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2

Rewrite

{\sqrt{10}}^{2}

${\sqrt{10}}^{2}$ as

10

$10$ .

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Step 2.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{10}

$\sqrt{10}$ as

10^{\frac{1}{2}}

$10^{\frac{1}{2}}$ .

\sqrt{\frac{{(10^{\frac{1}{2}})}^{2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}

$\sqrt{\frac{{(10^{\frac{1}{2}})}^{2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{\frac{10^{\frac{1}{2} \cdot 2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}

$\sqrt{\frac{10^{\frac{1}{2} \cdot 2}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{10^{\frac{2}{2}}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}

$\sqrt{\frac{10^{\frac{2}{2}}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2.4

Cancel the common factor of

2

$2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{\frac{10^{\frac{2}{2}}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{\frac{10^{1}}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{\frac{10}{10^{2}} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.3

Raise

$10$ to the power of

$2$ .

$\sqrt{\frac{10}{100} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.4

Cancel the common factor of

$10$ and

$100$ .

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Step 2.4.1

Factor

$10$ out of

$10$ .

$\sqrt{\frac{10 (1)}{100} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor

$10$ out of

$100$ .

$\sqrt{\frac{10 \cdot 1}{10 \cdot 10} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.4.2.2

Cancel the common factor.

$\sqrt{\frac{10 \cdot 1}{10 \cdot 10} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.4.2.3

Rewrite the expression.

$\sqrt{\frac{1}{10} + {(\frac{3 \sqrt{10}}{10})}^{2}}$

Step 2.5

Use the power rule

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

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Step 2.5.1

Apply the product rule to

$\frac{3 \sqrt{10}}{10}$ .

$\sqrt{\frac{1}{10} + \frac{{(3 \sqrt{10})}^{2}}{10^{2}}}$

Step 2.5.2

Apply the product rule to

$3 \sqrt{10}$ .

$\sqrt{\frac{1}{10} + \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{1}{10} + \frac{9 {\sqrt{10}}^{2}}{10^{2}}}$

Step 2.6.2

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 2.6.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{10} + \frac{9 {(10^{\frac{1}{2}})}^{2}}{10^{2}}}$

Step 2.6.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{1}{2} \cdot 2}}{10^{2}}}$

Step 2.6.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}}}$

Step 2.6.2.4

Cancel the common factor of

$2$ .

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Step 2.6.2.4.1

Cancel the common factor.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10^{1}}{10^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10}{10^{2}}}$

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

Raise

$10$ to the power of

$2$ .

$\sqrt{\frac{1}{10} + \frac{9 \cdot 10}{100}}$

Step 2.7.2

Multiply

$9$ by

$10$ .

$\sqrt{\frac{1}{10} + \frac{90}{100}}$

Step 2.7.3

Cancel the common factor of

$90$ and

$100$ .

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Step 2.7.3.1

Factor

$10$ out of

$90$ .

$\sqrt{\frac{1}{10} + \frac{10 (9)}{100}}$

Step 2.7.3.2

Cancel the common factors.

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Step 2.7.3.2.1

Factor

$10$ out of

$100$ .

$\sqrt{\frac{1}{10} + \frac{10 \cdot 9}{10 \cdot 10}}$

Step 2.7.3.2.2

Cancel the common factor.

$\sqrt{\frac{1}{10} + \frac{10 \cdot 9}{10 \cdot 10}}$

Step 2.7.3.2.3

Rewrite the expression.

$\sqrt{\frac{1}{10} + \frac{9}{10}}$

Step 2.7.4

Simplify the expression.

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Step 2.7.4.1

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 9}{10}}$

Step 2.7.4.2

Add

$1$ and

$9$ .

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

Step 2.7.4.3

Divide

$10$ by

$10$ .

$\sqrt{1}$

Step 2.7.4.4

Any root of

$1$ is

$1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore

$\cos (θ) = \frac{\frac{\sqrt{10}}{10}}{1}$ .

$\frac{\frac{\sqrt{10}}{10}}{1}$

Step 4

Divide

$\frac{\sqrt{10}}{10}$ by

$1$ .

$\cos (θ) = \frac{\sqrt{10}}{10}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{10}}{10} \approx 0.31622776$