Trigonometry Examples

$(\frac{5}{6}, \sqrt{\frac{11}{6}})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{5}{6}, \sqrt{\frac{11}{6}})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{5}{6}, 0)$ , and $(\frac{5}{6}, \sqrt{\frac{11}{6}})$ .

Opposite : $\sqrt{\frac{11}{6}}$

Adjacent : $\frac{5}{6}$

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 $\frac{5}{6}$ .

$\sqrt{\frac{5^{2}}{6^{2}} + {(\sqrt{\frac{11}{6}})}^{2}}$

Step 2.2

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

$\sqrt{\frac{25}{6^{2}} + {(\sqrt{\frac{11}{6}})}^{2}}$

Step 2.3

Raise $6$ to the power of $2$ .

$\sqrt{\frac{25}{36} + {(\sqrt{\frac{11}{6}})}^{2}}$

Step 2.4

Rewrite $\sqrt{\frac{11}{6}}$ as $\frac{\sqrt{11}}{\sqrt{6}}$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11}}{\sqrt{6}})}^{2}}$

Step 2.5

Multiply $\frac{\sqrt{11}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})}^{2}}$

Step 2.6

Combine and simplify the denominator.

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Step 2.6.1

Multiply $\frac{\sqrt{11}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{\sqrt{6} \sqrt{6}})}^{2}}$

Step 2.6.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}})}^{2}}$

Step 2.6.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}})}^{2}}$

Step 2.6.4

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

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{1 + 1}})}^{2}}$

Step 2.6.5

Add $1$ and $1$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{2}})}^{2}}$

Step 2.6.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}})}^{2}}$

Step 2.6.6.2

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

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}})}^{2}}$

Step 2.6.6.3

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

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{6^{\frac{2}{2}}})}^{2}}$

Step 2.6.6.4

Cancel the common factor of $2$ .

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Step 2.6.6.4.1

Cancel the common factor.

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{6^{\frac{2}{2}}})}^{2}}$

Step 2.6.6.4.2

Rewrite the expression.

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{6^{1}})}^{2}}$

Step 2.6.6.5

Evaluate the exponent.

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11} \sqrt{6}}{6})}^{2}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Combine using the product rule for radicals.

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{11 \cdot 6}}{6})}^{2}}$

Step 2.7.2

Multiply $11$ by $6$ .

$\sqrt{\frac{25}{36} + {(\frac{\sqrt{66}}{6})}^{2}}$

Step 2.8

Simplify terms.

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Step 2.8.1

Apply the product rule to $\frac{\sqrt{66}}{6}$ .

$\sqrt{\frac{25}{36} + \frac{{\sqrt{66}}^{2}}{6^{2}}}$

Step 2.8.2

Rewrite ${\sqrt{66}}^{2}$ as $66$ .

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Step 2.8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{66}$ as $66^{\frac{1}{2}}$ .

$\sqrt{\frac{25}{36} + \frac{{(66^{\frac{1}{2}})}^{2}}{6^{2}}}$

Step 2.8.2.2

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

$\sqrt{\frac{25}{36} + \frac{66^{\frac{1}{2} \cdot 2}}{6^{2}}}$

Step 2.8.2.3

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

$\sqrt{\frac{25}{36} + \frac{66^{\frac{2}{2}}}{6^{2}}}$

Step 2.8.2.4

Cancel the common factor of $2$ .

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Step 2.8.2.4.1

Cancel the common factor.

$\sqrt{\frac{25}{36} + \frac{66^{\frac{2}{2}}}{6^{2}}}$

Step 2.8.2.4.2

Rewrite the expression.

$\sqrt{\frac{25}{36} + \frac{66^{1}}{6^{2}}}$

Step 2.8.2.5

Evaluate the exponent.

$\sqrt{\frac{25}{36} + \frac{66}{6^{2}}}$

Step 2.8.3

Raise $6$ to the power of $2$ .

$\sqrt{\frac{25}{36} + \frac{66}{36}}$

Step 2.8.4

Cancel the common factor of $66$ and $36$ .

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Step 2.8.4.1

Factor $6$ out of $66$ .

$\sqrt{\frac{25}{36} + \frac{6 (11)}{36}}$

Step 2.8.4.2

Cancel the common factors.

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Step 2.8.4.2.1

Factor $6$ out of $36$ .

$\sqrt{\frac{25}{36} + \frac{6 \cdot 11}{6 \cdot 6}}$

Step 2.8.4.2.2

Cancel the common factor.

$\sqrt{\frac{25}{36} + \frac{6 \cdot 11}{6 \cdot 6}}$

Step 2.8.4.2.3

Rewrite the expression.

$\sqrt{\frac{25}{36} + \frac{11}{6}}$

Step 2.9

To write $\frac{11}{6}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\sqrt{\frac{25}{36} + \frac{11}{6} \cdot \frac{6}{6}}$

Step 2.10

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.10.1

Multiply $\frac{11}{6}$ by $\frac{6}{6}$ .

$\sqrt{\frac{25}{36} + \frac{11 \cdot 6}{6 \cdot 6}}$

Step 2.10.2

Multiply $6$ by $6$ .

$\sqrt{\frac{25}{36} + \frac{11 \cdot 6}{36}}$

Step 2.11

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + 11 \cdot 6}{36}}$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Multiply $11$ by $6$ .

$\sqrt{\frac{25 + 66}{36}}$

Step 2.12.2

Add $25$ and $66$ .

$\sqrt{\frac{91}{36}}$

Step 2.13

Rewrite $\sqrt{\frac{91}{36}}$ as $\frac{\sqrt{91}}{\sqrt{36}}$ .

$\frac{\sqrt{91}}{\sqrt{36}}$

Step 2.14

Simplify the denominator.

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Step 2.14.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{91}}{\sqrt{6^{2}}}$

Step 2.14.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{91}}{6}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{\frac{11}{6}}}{\frac{\sqrt{91}}{6}}$ .

$\frac{\sqrt{\frac{11}{6}}}{\frac{\sqrt{91}}{6}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \sqrt{\frac{11}{6}} (\frac{6}{\sqrt{91}})$

Step 4.2

Rewrite $\sqrt{\frac{11}{6}}$ as $\frac{\sqrt{11}}{\sqrt{6}}$ .

$\sin (θ) = \frac{\sqrt{11}}{\sqrt{6}} \cdot \frac{6}{\sqrt{91}}$

Step 4.3

Multiply $\frac{\sqrt{11}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sin (θ) = \frac{\sqrt{11}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{\sqrt{11}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{\sqrt{6} \sqrt{6}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{\sqrt{6} \sqrt{6}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{\sqrt{6} \sqrt{6}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.4

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

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{1 + 1}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{{\sqrt{6}}^{2}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6.2

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

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6.3

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

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6^{\frac{2}{2}}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6^{\frac{2}{2}}} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6} \cdot \frac{6}{\sqrt{91}}$

Step 4.4.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6} \cdot \frac{6}{\sqrt{91}}$

Step 4.5

Cancel the common factor of $6$ .

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Step 4.5.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{11} \sqrt{6}}{6} \cdot \frac{6}{\sqrt{91}}$

Step 4.5.2

Rewrite the expression.

$\sin (θ) = \sqrt{11} \sqrt{6} (\frac{1}{\sqrt{91}})$

Step 4.6

Combine using the product rule for radicals.

$\sin (θ) = \sqrt{11 \cdot 6} (\frac{1}{\sqrt{91}})$

Step 4.7

Multiply $11$ by $6$ .

$\sin (θ) = \sqrt{66} (\frac{1}{\sqrt{91}})$

Step 4.8

Combine $\sqrt{66}$ and $\frac{1}{\sqrt{91}}$ .

$\sin (θ) = \frac{\sqrt{66}}{\sqrt{91}}$

Step 4.9

Multiply $\frac{\sqrt{66}}{\sqrt{91}}$ by $\frac{\sqrt{91}}{\sqrt{91}}$ .

$\sin (θ) = \frac{\sqrt{66}}{\sqrt{91}} \cdot \frac{\sqrt{91}}{\sqrt{91}}$

Step 4.10

Combine and simplify the denominator.

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Step 4.10.1

Multiply $\frac{\sqrt{66}}{\sqrt{91}}$ by $\frac{\sqrt{91}}{\sqrt{91}}$ .

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{\sqrt{91} \sqrt{91}}$

Step 4.10.2

Raise $\sqrt{91}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{\sqrt{91} \sqrt{91}}$

Step 4.10.3

Raise $\sqrt{91}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{\sqrt{91} \sqrt{91}}$

Step 4.10.4

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

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{{\sqrt{91}}^{1 + 1}}$

Step 4.10.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{{\sqrt{91}}^{2}}$

Step 4.10.6

Rewrite ${\sqrt{91}}^{2}$ as $91$ .

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Step 4.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{91}$ as $91^{\frac{1}{2}}$ .

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{{(91^{\frac{1}{2}})}^{2}}$

Step 4.10.6.2

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

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{91^{\frac{1}{2} \cdot 2}}$

Step 4.10.6.3

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

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{91^{\frac{2}{2}}}$

Step 4.10.6.4

Cancel the common factor of $2$ .

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Step 4.10.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{91^{\frac{2}{2}}}$

Step 4.10.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{91}$

Step 4.10.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{66} \sqrt{91}}{91}$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Combine using the product rule for radicals.

$\sin (θ) = \frac{\sqrt{66 \cdot 91}}{91}$

Step 4.11.2

Multiply $66$ by $91$ .

$\sin (θ) = \frac{\sqrt{6006}}{91}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{6006}}{91} \approx 0.85163062$