Trigonometry Examples

$(\frac{\sqrt{21}}{5}, - \frac{2}{5})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{21}}{5}, - \frac{2}{5})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{21}}{5}, 0)$ , and $(\frac{\sqrt{21}}{5}, - \frac{2}{5})$ .

Opposite : $- \frac{2}{5}$

Adjacent : $\frac{\sqrt{21}}{5}$

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 $\frac{\sqrt{21}}{5}$ .

$\sqrt{\frac{{\sqrt{21}}^{2}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\sqrt{\frac{{(21^{\frac{1}{2}})}^{2}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2.2

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

$\sqrt{\frac{21^{\frac{1}{2} \cdot 2}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2.3

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

$\sqrt{\frac{21^{\frac{2}{2}}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{\frac{21^{\frac{2}{2}}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{\frac{21^{1}}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{\frac{21}{5^{2}} + {(- \frac{2}{5})}^{2}}$

Step 2.3

Raise $5$ to the power of $2$ .

$\sqrt{\frac{21}{25} + {(- \frac{2}{5})}^{2}}$

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 $- \frac{2}{5}$ .

$\sqrt{\frac{21}{25} + {(- 1)}^{2} {(\frac{2}{5})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{2}{5}$ .

$\sqrt{\frac{21}{25} + {(- 1)}^{2} \frac{2^{2}}{5^{2}}}$

Step 2.5

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

$\sqrt{\frac{21}{25} + 1 \frac{2^{2}}{5^{2}}}$

Step 2.6

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

$\sqrt{\frac{21}{25} + \frac{2^{2}}{5^{2}}}$

Step 2.7

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

$\sqrt{\frac{21}{25} + \frac{4}{5^{2}}}$

Step 2.8

Raise $5$ to the power of $2$ .

$\sqrt{\frac{21}{25} + \frac{4}{25}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{21 + 4}{25}}$

Step 2.10

Add $21$ and $4$ .

$\sqrt{\frac{25}{25}}$

Step 2.11

Divide $25$ by $25$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{2}{5}}{1}$ .

$\frac{- \frac{2}{5}}{1}$

Step 4

Divide $- \frac{2}{5}$ by $1$ .

$\sin (θ) = - \frac{2}{5}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{2}{5} \approx - 0.4$