Trigonometry Examples

$(5, - 4 \sqrt{5})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(5, - 4 \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(5, 0)$ , and $(5, - 4 \sqrt{5})$ .

Opposite : $- 4 \sqrt{5}$

Adjacent : $5$

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

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

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

Step 2.1.2

Apply the product rule to $- 4 \sqrt{5}$ .

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

Step 2.1.3

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

$\sqrt{25 + 16 {\sqrt{5}}^{2}}$

Step 2.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{25 + 16 {(5^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{25 + 16 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{25 + 16 \cdot 5^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{25 + 16 \cdot 5^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{25 + 16 \cdot 5^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{25 + 16 \cdot 5}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $16$ by $5$ .

$\sqrt{25 + 80}$

Step 2.3.2

Add $25$ and $80$ .

$\sqrt{105}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 4 \sqrt{5}}{\sqrt{105}}$ .

$\frac{- 4 \sqrt{5}}{\sqrt{105}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Combine $\sqrt{5}$ and $\sqrt{105}$ into a single radical.

$\sin (θ) = - 4 \sqrt{\frac{5}{105}}$

Step 4.2

Cancel the common factor of $5$ and $105$ .

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Step 4.2.1

Factor $5$ out of $5$ .

$\sin (θ) = - 4 \sqrt{\frac{5 (1)}{105}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $5$ out of $105$ .

$\sin (θ) = - 4 \sqrt{\frac{5 \cdot 1}{5 \cdot 21}}$

Step 4.2.2.2

Cancel the common factor.

$\sin (θ) = - 4 \sqrt{\frac{5 \cdot 1}{5 \cdot 21}}$

Step 4.2.2.3

Rewrite the expression.

$\sin (θ) = - 4 \sqrt{\frac{1}{21}}$

Step 4.3

Rewrite $\sqrt{\frac{1}{21}}$ as $\frac{\sqrt{1}}{\sqrt{21}}$ .

$\sin (θ) = - 4 \frac{\sqrt{1}}{\sqrt{21}}$

Step 4.4

Any root of $1$ is $1$ .

$\sin (θ) = - 4 \frac{1}{\sqrt{21}}$

Step 4.5

Multiply $\frac{1}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\sin (θ) = - 4 (\frac{1}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{1}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\sin (θ) = - 4 \frac{\sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 4.6.2

Raise $\sqrt{21}$ to the power of $1$ .

$\sin (θ) = - 4 \frac{\sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 4.6.3

Raise $\sqrt{21}$ to the power of $1$ .

$\sin (θ) = - 4 \frac{\sqrt{21}}{\sqrt{21} \sqrt{21}}$

Step 4.6.4

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

$\sin (θ) = - 4 \frac{\sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = - 4 \frac{\sqrt{21}}{{\sqrt{21}}^{2}}$

Step 4.6.6

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

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Step 4.6.6.1

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

$\sin (θ) = - 4 \frac{\sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = - 4 \frac{\sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = - 4 \frac{\sqrt{21}}{21^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$\sin (θ) = - 4 \frac{\sqrt{21}}{21^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = - 4 \frac{\sqrt{21}}{21}$

Step 4.6.6.5

Evaluate the exponent.

$\sin (θ) = - 4 \frac{\sqrt{21}}{21}$

Step 4.7

Combine $- 4$ and $\frac{\sqrt{21}}{21}$ .

$\sin (θ) = \frac{- 4 \sqrt{21}}{21}$

Step 4.8

Move the negative in front of the fraction.

$\sin (θ) = - \frac{4 \sqrt{21}}{21}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{4 \sqrt{21}}{21} \approx - 0.87287156$