Trigonometry Examples

$(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{2}}{5}, 0)$ , and $(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$ .

Opposite : $\frac{\sqrt{23}}{5}$

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

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

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{2}}{5})}^{2} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.1.2

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

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

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

$\sqrt{1 \frac{{\sqrt{2}}^{2}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.2.2

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

$\sqrt{\frac{{\sqrt{2}}^{2}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3

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

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Step 2.3.1

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

$\sqrt{\frac{{(2^{\frac{1}{2}})}^{2}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3.2

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

$\sqrt{\frac{2^{\frac{1}{2} \cdot 2}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3.3

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

$\sqrt{\frac{2^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{\frac{2^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{\frac{2^{1}}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{\frac{2}{5^{2}} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

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

$\sqrt{\frac{2}{25} + {(\frac{\sqrt{23}}{5})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{23}}{5}$ .

$\sqrt{\frac{2}{25} + \frac{{\sqrt{23}}^{2}}{5^{2}}}$

Step 2.5

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\sqrt{\frac{2}{25} + \frac{{(23^{\frac{1}{2}})}^{2}}{5^{2}}}$

Step 2.5.2

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

$\sqrt{\frac{2}{25} + \frac{23^{\frac{1}{2} \cdot 2}}{5^{2}}}$

Step 2.5.3

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

$\sqrt{\frac{2}{25} + \frac{23^{\frac{2}{2}}}{5^{2}}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\sqrt{\frac{2}{25} + \frac{23^{\frac{2}{2}}}{5^{2}}}$

Step 2.5.4.2

Rewrite the expression.

$\sqrt{\frac{2}{25} + \frac{23^{1}}{5^{2}}}$

Step 2.5.5

Evaluate the exponent.

$\sqrt{\frac{2}{25} + \frac{23}{5^{2}}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

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

$\sqrt{\frac{2}{25} + \frac{23}{25}}$

Step 2.6.2

Combine the numerators over the common denominator.

$\sqrt{\frac{2 + 23}{25}}$

Step 2.6.3

Add $2$ and $23$ .

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

Step 2.6.4

Divide $25$ by $25$ .

$\sqrt{1}$

Step 2.6.5

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{\sqrt{23}}{5}}{1}$ .

$\frac{\frac{\sqrt{23}}{5}}{1}$

Step 4

Divide $\frac{\sqrt{23}}{5}$ by $1$ .

$\sin (θ) = \frac{\sqrt{23}}{5}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{23}}{5} \approx 0.9591663$