Trigonometry Examples

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

Step 1

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

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

Adjacent : $\frac{1}{3}$

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 $\frac{1}{3}$ .

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

Step 2.2

One to any power is one.

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

Step 2.3

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

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

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

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

Step 2.4.2

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

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

$\sqrt{\frac{1}{9} + \frac{4 {\sqrt{2}}^{2}}{3^{2}}}$

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

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

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

Step 2.5.2.3

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

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

Step 2.5.2.4

Cancel the common factor of $2$ .

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Step 2.5.2.4.1

Cancel the common factor.

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

Step 2.5.2.4.2

Rewrite the expression.

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

Step 2.5.2.5

Evaluate the exponent.

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

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$\sqrt{\frac{1}{9} + \frac{4 \cdot 2}{9}}$

Step 2.6.2

Multiply $4$ by $2$ .

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

Step 2.6.3

Combine the numerators over the common denominator.

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

Step 2.6.4

Add $1$ and $8$ .

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

Step 2.6.5

Divide $9$ by $9$ .

$\sqrt{1}$

Step 2.6.6

Any root of $1$ is $1$ .

$1$

Step 3

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

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

Step 4

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

$\sin (θ) = \frac{2 \sqrt{2}}{3}$

Step 5

Approximate the result.

$\sin (θ) = \frac{2 \sqrt{2}}{3} \approx 0.94280904$