Trigonometry Examples

$(\frac{4}{7}, - \frac{\sqrt{33}}{7})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{4}{7}, - \frac{\sqrt{33}}{7})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{4}{7}, 0)$ , and $(\frac{4}{7}, - \frac{\sqrt{33}}{7})$ .

Opposite : $- \frac{\sqrt{33}}{7}$

Adjacent : $\frac{4}{7}$

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 $\frac{4}{7}$ .

$\sqrt{\frac{4^{2}}{7^{2}} + {(- \frac{\sqrt{33}}{7})}^{2}}$

Step 2.2

Raise $4$ to the power of $2$ .

$\sqrt{\frac{16}{7^{2}} + {(- \frac{\sqrt{33}}{7})}^{2}}$

Step 2.3

Raise $7$ to the power of $2$ .

$\sqrt{\frac{16}{49} + {(- \frac{\sqrt{33}}{7})}^{2}}$

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 $- \frac{\sqrt{33}}{7}$ .

$\sqrt{\frac{16}{49} + {(- 1)}^{2} {(\frac{\sqrt{33}}{7})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{33}}{7}$ .

$\sqrt{\frac{16}{49} + {(- 1)}^{2} \frac{{\sqrt{33}}^{2}}{7^{2}}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

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

$\sqrt{\frac{16}{49} + 1 \frac{{\sqrt{33}}^{2}}{7^{2}}}$

Step 2.5.2

Multiply $\frac{{\sqrt{33}}^{2}}{7^{2}}$ by $1$ .

$\sqrt{\frac{16}{49} + \frac{{\sqrt{33}}^{2}}{7^{2}}}$

Step 2.6

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\sqrt{\frac{16}{49} + \frac{{(33^{\frac{1}{2}})}^{2}}{7^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{16}{49} + \frac{33^{\frac{1}{2} \cdot 2}}{7^{2}}}$

Step 2.6.3

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

$\sqrt{\frac{16}{49} + \frac{33^{\frac{2}{2}}}{7^{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{16}{49} + \frac{33^{\frac{2}{2}}}{7^{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{16}{49} + \frac{33^{1}}{7^{2}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{16}{49} + \frac{33}{7^{2}}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Raise $7$ to the power of $2$ .

$\sqrt{\frac{16}{49} + \frac{33}{49}}$

Step 2.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{16 + 33}{49}}$

Step 2.7.3

Add $16$ and $33$ .

$\sqrt{\frac{49}{49}}$

Step 2.7.4

Divide $49$ by $49$ .

$\sqrt{1}$

Step 2.7.5

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{\sqrt{33}}{7}}{1}$ .

$\frac{- \frac{\sqrt{33}}{7}}{1}$

Step 4

Divide $- \frac{\sqrt{33}}{7}$ by $1$ .

$\sin (θ) = - \frac{\sqrt{33}}{7}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{\sqrt{33}}{7} \approx - 0.8206518$