Trigonometry Examples

$(\frac{5}{8}, - \frac{3}{8})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{5}{8}, - \frac{3}{8})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{5}{8}, 0)$ , and $(\frac{5}{8}, - \frac{3}{8})$ .

Opposite : $- \frac{3}{8}$

Adjacent : $\frac{5}{8}$

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 $\frac{5}{8}$ .

$\sqrt{\frac{5^{2}}{8^{2}} + {(- \frac{3}{8})}^{2}}$

Step 2.2

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

$\sqrt{\frac{25}{8^{2}} + {(- \frac{3}{8})}^{2}}$

Step 2.3

Raise $8$ to the power of $2$ .

$\sqrt{\frac{25}{64} + {(- \frac{3}{8})}^{2}}$

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 $- \frac{3}{8}$ .

$\sqrt{\frac{25}{64} + {(- 1)}^{2} {(\frac{3}{8})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{3}{8}$ .

$\sqrt{\frac{25}{64} + {(- 1)}^{2} \frac{3^{2}}{8^{2}}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

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

$\sqrt{\frac{25}{64} + 1 \frac{3^{2}}{8^{2}}}$

Step 2.5.2

Multiply $\frac{3^{2}}{8^{2}}$ by $1$ .

$\sqrt{\frac{25}{64} + \frac{3^{2}}{8^{2}}}$

Step 2.5.3

Raise $3$ to the power of $2$ .

$\sqrt{\frac{25}{64} + \frac{9}{8^{2}}}$

Step 2.5.4

Raise $8$ to the power of $2$ .

$\sqrt{\frac{25}{64} + \frac{9}{64}}$

Step 2.5.5

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + 9}{64}}$

Step 2.5.6

Add $25$ and $9$ .

$\sqrt{\frac{34}{64}}$

Step 2.6

Cancel the common factor of $34$ and $64$ .

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Step 2.6.1

Factor $2$ out of $34$ .

$\sqrt{\frac{2 (17)}{64}}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $64$ .

$\sqrt{\frac{2 \cdot 17}{2 \cdot 32}}$

Step 2.6.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 17}{2 \cdot 32}}$

Step 2.6.2.3

Rewrite the expression.

$\sqrt{\frac{17}{32}}$

Step 2.7

Rewrite $\sqrt{\frac{17}{32}}$ as $\frac{\sqrt{17}}{\sqrt{32}}$ .

$\frac{\sqrt{17}}{\sqrt{32}}$

Step 2.8

Simplify the denominator.

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Step 2.8.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.8.1.1

Factor $16$ out of $32$ .

$\frac{\sqrt{17}}{\sqrt{16 (2)}}$

Step 2.8.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{17}}{\sqrt{4^{2} \cdot 2}}$

Step 2.8.2

Pull terms out from under the radical.

$\frac{\sqrt{17}}{4 \sqrt{2}}$

Step 2.9

Multiply $\frac{\sqrt{17}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{17}}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.10

Combine and simplify the denominator.

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Step 2.10.1

Multiply $\frac{\sqrt{17}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{17} \sqrt{2}}{4 \sqrt{2} \sqrt{2}}$

Step 2.10.2

Move $\sqrt{2}$ .

$\frac{\sqrt{17} \sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

Step 2.10.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{17} \sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 2.10.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{17} \sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 2.10.5

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

$\frac{\sqrt{17} \sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Step 2.10.6

Add $1$ and $1$ .

$\frac{\sqrt{17} \sqrt{2}}{4 {\sqrt{2}}^{2}}$

Step 2.10.7

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

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Step 2.10.7.1

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

$\frac{\sqrt{17} \sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

Step 2.10.7.2

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

$\frac{\sqrt{17} \sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.10.7.3

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

$\frac{\sqrt{17} \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 2.10.7.4

Cancel the common factor of $2$ .

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Step 2.10.7.4.1

Cancel the common factor.

$\frac{\sqrt{17} \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Step 2.10.7.4.2

Rewrite the expression.

$\frac{\sqrt{17} \sqrt{2}}{4 \cdot 2^{1}}$

Step 2.10.7.5

Evaluate the exponent.

$\frac{\sqrt{17} \sqrt{2}}{4 \cdot 2}$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Combine using the product rule for radicals.

$\frac{\sqrt{17 \cdot 2}}{4 \cdot 2}$

Step 2.11.2

Multiply $17$ by $2$ .

$\frac{\sqrt{34}}{4 \cdot 2}$

Step 2.12

Multiply $4$ by $2$ .

$\frac{\sqrt{34}}{8}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{3}{8}}{\frac{\sqrt{34}}{8}}$ .

$\frac{- \frac{3}{8}}{\frac{\sqrt{34}}{8}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = - \frac{3}{8} \cdot \frac{8}{\sqrt{34}}$

Step 4.2

Cancel the common factor of $8$ .

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Step 4.2.1

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$\sin (θ) = \frac{- 3}{8} \cdot \frac{8}{\sqrt{34}}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{- 3}{8} \cdot \frac{8}{\sqrt{34}}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = - 3 \frac{1}{\sqrt{34}}$

Step 4.3

Combine $- 3$ and $\frac{1}{\sqrt{34}}$ .

$\sin (θ) = \frac{- 3}{\sqrt{34}}$

Step 4.4

Move the negative in front of the fraction.

$\sin (θ) = - \frac{3}{\sqrt{34}}$

Step 4.5

Multiply $\frac{3}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sin (θ) = - (\frac{3}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{3}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sin (θ) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.6.2

Raise $\sqrt{34}$ to the power of $1$ .

$\sin (θ) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.6.3

Raise $\sqrt{34}$ to the power of $1$ .

$\sin (θ) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 4.6.4

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

$\sin (θ) = - \frac{3 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{3 \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{3 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = - \frac{3 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = - \frac{3 \sqrt{34}}{34^{\frac{2}{2}}}$

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.

$\sin (θ) = - \frac{3 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{3 \sqrt{34}}{34}$

Step 4.6.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{3 \sqrt{34}}{34}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{3 \sqrt{34}}{34} \approx - 0.51449575$