Trigonometry Examples

(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})

$(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})$

Step 1

To find the

sec (θ)

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

(0, 0)

$(0, 0)$ and

(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})

$(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- \frac{2}{7}, 0)

$(- \frac{2}{7}, 0)$ , and

(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})

$(- \frac{2}{7}, \frac{3 \sqrt{5}}{7})$ .

Opposite :

\frac{3 \sqrt{5}}{7}

$\frac{3 \sqrt{5}}{7}$

Adjacent :

- \frac{2}{7}

$- \frac{2}{7}$

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}{7}

$- \frac{2}{7}$ .

\sqrt{{(- 1)}^{2} {(\frac{2}{7})}^{2} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{{(- 1)}^{2} {(\frac{2}{7})}^{2} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.1.2

Apply the product rule to

\frac{2}{7}

$\frac{2}{7}$ .

\sqrt{{(- 1)}^{2} \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{{(- 1)}^{2} \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

\sqrt{{(- 1)}^{2} \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{{(- 1)}^{2} \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

\sqrt{1 \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{1 \frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.3

Multiply

\frac{2^{2}}{7^{2}}

$\frac{2^{2}}{7^{2}}$ by

1

$1$ .

\sqrt{\frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{\frac{2^{2}}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.4

Raise

2

$2$ to the power of

2

$2$ .

\sqrt{\frac{4}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{\frac{4}{7^{2}} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.5

Raise

7

$7$ to the power of

2

$2$ .

\sqrt{\frac{4}{49} + {(\frac{3 \sqrt{5}}{7})}^{2}}

$\sqrt{\frac{4}{49} + {(\frac{3 \sqrt{5}}{7})}^{2}}$

Step 2.6

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 2.6.1

Apply the product rule to

\frac{3 \sqrt{5}}{7}

$\frac{3 \sqrt{5}}{7}$ .

\sqrt{\frac{4}{49} + \frac{{(3 \sqrt{5})}^{2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{{(3 \sqrt{5})}^{2}}{7^{2}}}$

Step 2.6.2

Apply the product rule to

3 \sqrt{5}

$3 \sqrt{5}$ .

\sqrt{\frac{4}{49} + \frac{3^{2} {\sqrt{5}}^{2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{3^{2} {\sqrt{5}}^{2}}{7^{2}}}$

\sqrt{\frac{4}{49} + \frac{3^{2} {\sqrt{5}}^{2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{3^{2} {\sqrt{5}}^{2}}{7^{2}}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{\frac{4}{49} + \frac{9 {\sqrt{5}}^{2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{9 {\sqrt{5}}^{2}}{7^{2}}}$

Step 2.7.2

Rewrite

{\sqrt{5}}^{2}

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

5

$5$ .

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Step 2.7.2.1

Use

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

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

\sqrt{5}

$\sqrt{5}$ as

5^{\frac{1}{2}}

$5^{\frac{1}{2}}$ .

\sqrt{\frac{4}{49} + \frac{9 {(5^{\frac{1}{2}})}^{2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{9 {(5^{\frac{1}{2}})}^{2}}{7^{2}}}$

Step 2.7.2.2

Apply the power rule and multiply exponents,

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

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

\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{\frac{1}{2} \cdot 2}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{\frac{1}{2} \cdot 2}}{7^{2}}}$

Step 2.7.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{\frac{2}{2}}}{7^{2}}}

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{\frac{2}{2}}}{7^{2}}}$

Step 2.7.2.4

Cancel the common factor of

2

$2$ .

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Step 2.7.2.4.1

Cancel the common factor.

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{\frac{2}{2}}}{7^{2}}}$

Step 2.7.2.4.2

Rewrite the expression.

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5^{1}}{7^{2}}}$

Step 2.7.2.5

Evaluate the exponent.

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5}{7^{2}}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Raise

$7$ to the power of

$2$ .

$\sqrt{\frac{4}{49} + \frac{9 \cdot 5}{49}}$

Step 2.8.2

Multiply

$9$ by

$5$ .

$\sqrt{\frac{4}{49} + \frac{45}{49}}$

Step 2.8.3

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + 45}{49}}$

Step 2.8.4

Add

$4$ and

$45$ .

$\sqrt{\frac{49}{49}}$

Step 2.8.5

Divide

$49$ by

$49$ .

$\sqrt{1}$

Step 2.8.6

Any root of

$1$ is

$1$ .

$1$

Step 3

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

$\sec (θ) = \frac{1}{- \frac{2}{7}}$ .

$\frac{1}{- \frac{2}{7}}$

Step 4

Simplify

$\sec (θ)$ .

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Step 4.1

Cancel the common factor of

$1$ and

$- 1$ .

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Step 4.1.1

Rewrite

$1$ as

$- 1 (- 1)$ .

$\sec (θ) = \frac{- 1 \cdot - 1}{- \frac{2}{7}}$

Step 4.1.2

Move the negative in front of the fraction.

$\sec (θ) = - \frac{1}{\frac{2}{7}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = - (1 (\frac{7}{2}))$

Step 4.3

Multiply

$\frac{7}{2}$ by

$1$ .

$\sec (θ) = - \frac{7}{2}$

Step 5

Approximate the result.

$\sec (θ) = - \frac{7}{2} \approx - 3.5$