Trigonometry Examples

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

Step 1

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

Opposite : $- \frac{\sqrt{21}}{5}$

Adjacent : $\frac{2}{5}$

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

$\sqrt{\frac{4}{5^{2}} + {(- \frac{\sqrt{21}}{5})}^{2}}$

Step 2.3

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

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

Apply the product rule to $- \frac{\sqrt{21}}{5}$ .

$\sqrt{\frac{4}{25} + {(- 1)}^{2} {(\frac{\sqrt{21}}{5})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{21}}{5}$ .

$\sqrt{\frac{4}{25} + {(- 1)}^{2} \frac{{\sqrt{21}}^{2}}{5^{2}}}$

Step 2.5

Simplify the expression.

Tap for more steps...

Step 2.5.1

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

$\sqrt{\frac{4}{25} + 1 \frac{{\sqrt{21}}^{2}}{5^{2}}}$

Step 2.5.2

Multiply $\frac{{\sqrt{21}}^{2}}{5^{2}}$ by $1$ .

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

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\sqrt{\frac{4}{25} + \frac{{(21^{\frac{1}{2}})}^{2}}{5^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{4}{25} + \frac{21^{\frac{1}{2} \cdot 2}}{5^{2}}}$

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{4}{25} + \frac{21^{1}}{5^{2}}}$

Step 2.6.5

Evaluate the exponent.

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

Step 2.7

Simplify the expression.

Tap for more steps...

Step 2.7.1

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

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

Step 2.7.2

Combine the numerators over the common denominator.

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

Step 2.7.3

Add $4$ and $21$ .

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

Step 2.7.4

Divide $25$ by $25$ .

$\sqrt{1}$

Step 2.7.5

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{\sqrt{21}}{5}}{1}$ .

$\frac{- \frac{\sqrt{21}}{5}}{1}$

Step 4

Divide $- \frac{\sqrt{21}}{5}$ by $1$ .

$\sin (θ) = - \frac{\sqrt{21}}{5}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{\sqrt{21}}{5} \approx - 0.91651513$