Trigonometry Examples

$(- \sqrt{33}, - 60)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \sqrt{33}, - 60)$ , draw the triangle between the three points $(0, 0)$ , $(- \sqrt{33}, 0)$ , and $(- \sqrt{33}, - 60)$ .

Opposite : $- 60$

Adjacent : $- \sqrt{33}$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

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

$\sqrt{{(- 1)}^{2} {\sqrt{33}}^{2} + {(- 60)}^{2}}$

Step 2.1.2

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

$\sqrt{1 {\sqrt{33}}^{2} + {(- 60)}^{2}}$

Step 2.1.3

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

$\sqrt{{\sqrt{33}}^{2} + {(- 60)}^{2}}$

Step 2.2

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

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Step 2.2.1

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

$\sqrt{{(33^{\frac{1}{2}})}^{2} + {(- 60)}^{2}}$

Step 2.2.2

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

$\sqrt{33^{\frac{1}{2} \cdot 2} + {(- 60)}^{2}}$

Step 2.2.3

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

$\sqrt{33^{\frac{2}{2}} + {(- 60)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{33^{\frac{2}{2}} + {(- 60)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{33^{1} + {(- 60)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{33 + {(- 60)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $- 60$ to the power of $2$ .

$\sqrt{33 + 3600}$

Step 2.3.2

Add $33$ and $3600$ .

$\sqrt{3633}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 60}{\sqrt{3633}}$ .

$\frac{- 60}{\sqrt{3633}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{60}{\sqrt{3633}}$

Step 4.2

Multiply $\frac{60}{\sqrt{3633}}$ by $\frac{\sqrt{3633}}{\sqrt{3633}}$ .

$\sin (θ) = - (\frac{60}{\sqrt{3633}} \cdot \frac{\sqrt{3633}}{\sqrt{3633}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{60}{\sqrt{3633}}$ by $\frac{\sqrt{3633}}{\sqrt{3633}}$ .

$\sin (θ) = - \frac{60 \sqrt{3633}}{\sqrt{3633} \sqrt{3633}}$

Step 4.3.2

Raise $\sqrt{3633}$ to the power of $1$ .

$\sin (θ) = - \frac{60 \sqrt{3633}}{\sqrt{3633} \sqrt{3633}}$

Step 4.3.3

Raise $\sqrt{3633}$ to the power of $1$ .

$\sin (θ) = - \frac{60 \sqrt{3633}}{\sqrt{3633} \sqrt{3633}}$

Step 4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = - \frac{60 \sqrt{3633}}{{\sqrt{3633}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{60 \sqrt{3633}}{{\sqrt{3633}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{3633}}^{2}$ as $3633$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3633}$ as $3633^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{60 \sqrt{3633}}{{(3633^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\sin (θ) = - \frac{60 \sqrt{3633}}{3633^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\sin (θ) = - \frac{60 \sqrt{3633}}{3633^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{60 \sqrt{3633}}{3633^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{60 \sqrt{3633}}{3633}$

Step 4.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{60 \sqrt{3633}}{3633}$

Step 4.4

Cancel the common factor of $60$ and $3633$ .

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Step 4.4.1

Factor $3$ out of $60 \sqrt{3633}$ .

$\sin (θ) = - \frac{3 (20 \sqrt{3633})}{3633}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $3$ out of $3633$ .

$\sin (θ) = - \frac{3 (20 \sqrt{3633})}{3 (1211)}$

Step 4.4.2.2

Cancel the common factor.

$\sin (θ) = - \frac{3 (20 \sqrt{3633})}{3 \cdot 1211}$

Step 4.4.2.3

Rewrite the expression.

$\sin (θ) = - \frac{20 \sqrt{3633}}{1211}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{20 \sqrt{3633}}{1211} \approx - 0.99544793$