Trigonometry Examples

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

Step 1

To find the $\csc (θ)$ 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 2.3.3

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

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

Step 2.3.4

One to any power is one.

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

Step 2.3.5

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

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

Step 2.3.6

Combine the numerators over the common denominator.

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

Step 2.3.7

Add $8$ and $1$ .

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

Step 2.3.8

Divide $9$ by $9$ .

$\sqrt{1}$

Step 2.3.9

Any root of $1$ is $1$ .

$1$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{\frac{1}{3}}$ .

$\frac{1}{\frac{1}{3}}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = 1 \cdot 3$

Step 4.2

Multiply $3$ by $1$ .

$\csc (θ) = 3$