Trigonometry Examples

$(\frac{2 \sqrt{2}}{3}, - \frac{1}{3})$

Step 1

To find the $\sec (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{2 \sqrt{2}}{3}, - \frac{1}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{2 \sqrt{2}}{3}, 0)$ , and $(\frac{2 \sqrt{2}}{3}, - \frac{1}{3})$ .

Opposite : $- \frac{1}{3}$

Adjacent : $\frac{2 \sqrt{2}}{3}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Apply the product rule to $\frac{2 \sqrt{2}}{3}$ .

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

Step 2.1.2

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

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\sqrt{\frac{4 {\sqrt{2}}^{2}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2

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

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Step 2.2.2.1

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

$\sqrt{\frac{4 {(2^{\frac{1}{2}})}^{2}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2.2

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

$\sqrt{\frac{4 \cdot 2^{\frac{1}{2} \cdot 2}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2.3

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

$\sqrt{\frac{4 \cdot 2^{\frac{2}{2}}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{4 \cdot 2^{\frac{2}{2}}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{4 \cdot 2^{1}}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.2.2.5

Evaluate the exponent.

$\sqrt{\frac{4 \cdot 2}{3^{2}} + {(- \frac{1}{3})}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{4 \cdot 2}{9} + {(- \frac{1}{3})}^{2}}$

Step 2.3.2

Multiply $4$ by $2$ .

$\sqrt{\frac{8}{9} + {(- \frac{1}{3})}^{2}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{1}{3}$ .

$\sqrt{\frac{8}{9} + {(- 1)}^{2} {(\frac{1}{3})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{\frac{8}{9} + {(- 1)}^{2} \frac{1^{2}}{3^{2}}}$

Step 2.5

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

$\sqrt{\frac{8}{9} + 1 \frac{1^{2}}{3^{2}}}$

Step 2.6

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{8}{9} + \frac{1^{2}}{3^{2}}}$

Step 2.7

One to any power is one.

$\sqrt{\frac{8}{9} + \frac{1}{3^{2}}}$

Step 2.8

Raise $3$ to the power of $2$ .

$\sqrt{\frac{8}{9} + \frac{1}{9}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{8 + 1}{9}}$

Step 2.10

Add $8$ and $1$ .

$\sqrt{\frac{9}{9}}$

Step 2.11

Divide $9$ by $9$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\sec (θ) = \frac{Hypotenuse}{Adjacent}$ therefore $\sec (θ) = \frac{1}{\frac{2 \sqrt{2}}{3}}$ .

$\frac{1}{\frac{2 \sqrt{2}}{3}}$

Step 4

Simplify $\sec (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = 1 (\frac{3}{2 \sqrt{2}})$

Step 4.2

Multiply $\frac{3}{2 \sqrt{2}}$ by $1$ .

$\sec (θ) = \frac{3}{2 \sqrt{2}}$

Step 4.3

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (θ) = \frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (θ) = \frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 4.4.2

Move $\sqrt{2}$ .

$\sec (θ) = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.4.3

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.4.4

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 4.4.5

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 4.4.6

Add $1$ and $1$ .

$\sec (θ) = \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 4.4.7

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

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Step 4.4.7.1

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 4.4.7.2

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.4.7.3

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

$\sec (θ) = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.4.7.4

Cancel the common factor of $2$ .

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Step 4.4.7.4.1

Cancel the common factor.

$\sec (θ) = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 4.4.7.4.2

Rewrite the expression.

$\sec (θ) = \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 4.4.7.5

Evaluate the exponent.

$\sec (θ) = \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 4.5

Multiply $2$ by $2$ .

$\sec (θ) = \frac{3 \sqrt{2}}{4}$

Step 5

Approximate the result.

$\sec (θ) = \frac{3 \sqrt{2}}{4} \approx 1.06066017$