Trigonometry Examples

(\frac{2 \sqrt{30}}{11}, \frac{1}{11})

$(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$

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{2 \sqrt{30}}{11}, \frac{1}{11})

$(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(\frac{2 \sqrt{30}}{11}, 0)

$(\frac{2 \sqrt{30}}{11}, 0)$ , and

(\frac{2 \sqrt{30}}{11}, \frac{1}{11})

$(\frac{2 \sqrt{30}}{11}, \frac{1}{11})$ .

Opposite :

\frac{1}{11}

$\frac{1}{11}$

Adjacent :

\frac{2 \sqrt{30}}{11}

$\frac{2 \sqrt{30}}{11}$

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

Use the power rule

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

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

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Step 2.1.1

Apply the product rule to

\frac{2 \sqrt{30}}{11}

$\frac{2 \sqrt{30}}{11}$ .

\sqrt{\frac{{(2 \sqrt{30})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{{(2 \sqrt{30})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.1.2

Apply the product rule to

2 \sqrt{30}

$2 \sqrt{30}$ .

\sqrt{\frac{2^{2} {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{2^{2} {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

\sqrt{\frac{2^{2} {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{2^{2} {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Raise

2

$2$ to the power of

2

$2$ .

\sqrt{\frac{4 {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{4 {\sqrt{30}}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2

Rewrite

{\sqrt{30}}^{2}

${\sqrt{30}}^{2}$ as

30

$30$ .

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Step 2.2.2.1

Use

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

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

\sqrt{30}

$\sqrt{30}$ as

30^{\frac{1}{2}}

$30^{\frac{1}{2}}$ .

\sqrt{\frac{4 {(30^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{4 {(30^{\frac{1}{2}})}^{2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.2

Apply the power rule and multiply exponents,

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

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

\sqrt{\frac{4 \cdot 30^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{4 \cdot 30^{\frac{1}{2} \cdot 2}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{4 \cdot 30^{\frac{2}{2}}}{11^{2}} + {(\frac{1}{11})}^{2}}

$\sqrt{\frac{4 \cdot 30^{\frac{2}{2}}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.4

Cancel the common factor of

2

$2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{4 \cdot 30^{\frac{2}{2}}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{4 \cdot 30^{1}}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.2.2.5

Evaluate the exponent.

$\sqrt{\frac{4 \cdot 30}{11^{2}} + {(\frac{1}{11})}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise

$11$ to the power of

$2$ .

$\sqrt{\frac{4 \cdot 30}{121} + {(\frac{1}{11})}^{2}}$

Step 2.3.2

Multiply

$4$ by

$30$ .

$\sqrt{\frac{120}{121} + {(\frac{1}{11})}^{2}}$

Step 2.3.3

Apply the product rule to

$\frac{1}{11}$ .

$\sqrt{\frac{120}{121} + \frac{1^{2}}{11^{2}}}$

Step 2.3.4

One to any power is one.

$\sqrt{\frac{120}{121} + \frac{1}{11^{2}}}$

Step 2.3.5

Raise

$11$ to the power of

$2$ .

$\sqrt{\frac{120}{121} + \frac{1}{121}}$

Step 2.3.6

Combine the numerators over the common denominator.

$\sqrt{\frac{120 + 1}{121}}$

Step 2.3.7

Add

$120$ and

$1$ .

$\sqrt{\frac{121}{121}}$

Step 2.3.8

Divide

$121$ by

$121$ .

$\sqrt{1}$

Step 2.3.9

Any root of

$1$ is

$1$ .

$1$

Step 3

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

$\cos (θ) = \frac{\frac{2 \sqrt{30}}{11}}{1}$ .

$\frac{\frac{2 \sqrt{30}}{11}}{1}$

Step 4

Divide

$\frac{2 \sqrt{30}}{11}$ by

$1$ .

$\cos (θ) = \frac{2 \sqrt{30}}{11}$

Step 5

Approximate the result.

$\cos (θ) = \frac{2 \sqrt{30}}{11} \approx 0.99585919$