Trigonometry Examples

$(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{3}}{3}, 0)$ , and $(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$ .

Opposite : $\frac{\sqrt{6}}{3}$

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

Step 2

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

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Step 2.1

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

$\sqrt{\frac{{\sqrt{3}}^{2}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{\frac{{(3^{\frac{1}{2}})}^{2}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2.2

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

$\sqrt{\frac{3^{\frac{1}{2} \cdot 2}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2.3

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

$\sqrt{\frac{3^{\frac{2}{2}}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{\frac{3^{\frac{2}{2}}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{\frac{3^{1}}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{\frac{3}{3^{2}} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.3

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

$\sqrt{\frac{3}{9} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.4

Cancel the common factor of $3$ and $9$ .

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Step 2.4.1

Factor $3$ out of $3$ .

$\sqrt{\frac{3 (1)}{9} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $9$ .

$\sqrt{\frac{3 \cdot 1}{3 \cdot 3} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.4.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 1}{3 \cdot 3} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.4.2.3

Rewrite the expression.

$\sqrt{\frac{1}{3} + {(\frac{\sqrt{6}}{3})}^{2}}$

Step 2.5

Apply the product rule to $\frac{\sqrt{6}}{3}$ .

$\sqrt{\frac{1}{3} + \frac{{\sqrt{6}}^{2}}{3^{2}}}$

Step 2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{3} + \frac{{(6^{\frac{1}{2}})}^{2}}{3^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{1}{3} + \frac{6^{\frac{1}{2} \cdot 2}}{3^{2}}}$

Step 2.6.3

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

$\sqrt{\frac{1}{3} + \frac{6^{\frac{2}{2}}}{3^{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{3} + \frac{6^{\frac{2}{2}}}{3^{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{3} + \frac{6^{1}}{3^{2}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1}{3} + \frac{6}{3^{2}}}$

Step 2.7

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

$\sqrt{\frac{1}{3} + \frac{6}{9}}$

Step 2.8

Cancel the common factor of $6$ and $9$ .

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Step 2.8.1

Factor $3$ out of $6$ .

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

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $3$ out of $9$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Combine the numerators over the common denominator.

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

Step 2.9.2

Add $1$ and $2$ .

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

Step 2.9.3

Divide $3$ by $3$ .

$\sqrt{1}$

Step 2.9.4

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{\sqrt{6}}{3}}{1}$ .

$\frac{\frac{\sqrt{6}}{3}}{1}$

Step 4

Divide $\frac{\sqrt{6}}{3}$ by $1$ .

$\sin (θ) = \frac{\sqrt{6}}{3}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{6}}{3} \approx 0.81649658$