Trigonometry Examples

$(- \frac{1}{2}, \sqrt{\frac{3}{2}})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{2}, \sqrt{\frac{3}{2}})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{2}, 0)$ , and $(- \frac{1}{2}, \sqrt{\frac{3}{2}})$ .

Opposite : $\sqrt{\frac{3}{2}}$

Adjacent : $- \frac{1}{2}$

Step 2

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

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 2.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.2

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

$\sqrt{1 \frac{1^{2}}{2^{2}} + {(\sqrt{\frac{3}{2}})}^{2}}$

Step 2.3

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

$\sqrt{\frac{1^{2}}{2^{2}} + {(\sqrt{\frac{3}{2}})}^{2}}$

Step 2.4

One to any power is one.

$\sqrt{\frac{1}{2^{2}} + {(\sqrt{\frac{3}{2}})}^{2}}$

Step 2.5

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

$\sqrt{\frac{1}{4} + {(\sqrt{\frac{3}{2}})}^{2}}$

Step 2.6

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3}}{\sqrt{2}})}^{2}}$

Step 2.7

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}}$

Step 2.8

Combine and simplify the denominator.

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Step 2.8.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}}$

Step 2.8.2

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}}$

Step 2.8.3

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}}$

Step 2.8.4

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}}$

Step 2.8.5

Add $1$ and $1$ .

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}}$

Step 2.8.6

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

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Step 2.8.6.1

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}}$

Step 2.8.6.2

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}}$

Step 2.8.6.3

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}$

Step 2.8.6.4

Cancel the common factor of $2$ .

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Step 2.8.6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}$

Step 2.8.6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{2^{1}})}^{2}}$

Step 2.8.6.5

Evaluate the exponent.

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3} \sqrt{2}}{2})}^{2}}$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Combine using the product rule for radicals.

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3 \cdot 2}}{2})}^{2}}$

Step 2.9.2

Multiply $3$ by $2$ .

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{6}}{2})}^{2}}$

Step 2.10

Simplify terms.

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Step 2.10.1

Apply the product rule to $\frac{\sqrt{6}}{2}$ .

$\sqrt{\frac{1}{4} + \frac{{\sqrt{6}}^{2}}{2^{2}}}$

Step 2.10.2

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

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Step 2.10.2.1

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

$\sqrt{\frac{1}{4} + \frac{{(6^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.10.2.2

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

$\sqrt{\frac{1}{4} + \frac{6^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.10.2.3

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

$\sqrt{\frac{1}{4} + \frac{6^{\frac{2}{2}}}{2^{2}}}$

Step 2.10.2.4

Cancel the common factor of $2$ .

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Step 2.10.2.4.1

Cancel the common factor.

$\sqrt{\frac{1}{4} + \frac{6^{\frac{2}{2}}}{2^{2}}}$

Step 2.10.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{4} + \frac{6^{1}}{2^{2}}}$

Step 2.10.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{4} + \frac{6}{2^{2}}}$

Step 2.10.3

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

$\sqrt{\frac{1}{4} + \frac{6}{4}}$

Step 2.10.4

Cancel the common factor of $6$ and $4$ .

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Step 2.10.4.1

Factor $2$ out of $6$ .

$\sqrt{\frac{1}{4} + \frac{2 (3)}{4}}$

Step 2.10.4.2

Cancel the common factors.

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Step 2.10.4.2.1

Factor $2$ out of $4$ .

$\sqrt{\frac{1}{4} + \frac{2 \cdot 3}{2 \cdot 2}}$

Step 2.10.4.2.2

Cancel the common factor.

$\sqrt{\frac{1}{4} + \frac{2 \cdot 3}{2 \cdot 2}}$

Step 2.10.4.2.3

Rewrite the expression.

$\sqrt{\frac{1}{4} + \frac{3}{2}}$

Step 2.11

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{1}{4} + \frac{3}{2} \cdot \frac{2}{2}}$

Step 2.12

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.12.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\sqrt{\frac{1}{4} + \frac{3 \cdot 2}{2 \cdot 2}}$

Step 2.12.2

Multiply $2$ by $2$ .

$\sqrt{\frac{1}{4} + \frac{3 \cdot 2}{4}}$

Step 2.13

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 3 \cdot 2}{4}}$

Step 2.14

Simplify the numerator.

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Step 2.14.1

Multiply $3$ by $2$ .

$\sqrt{\frac{1 + 6}{4}}$

Step 2.14.2

Add $1$ and $6$ .

$\sqrt{\frac{7}{4}}$

Step 2.15

Rewrite $\sqrt{\frac{7}{4}}$ as $\frac{\sqrt{7}}{\sqrt{4}}$ .

$\frac{\sqrt{7}}{\sqrt{4}}$

Step 2.16

Simplify the denominator.

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Step 2.16.1

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

$\frac{\sqrt{7}}{\sqrt{2^{2}}}$

Step 2.16.2

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

$\frac{\sqrt{7}}{2}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{\frac{3}{2}}}{\frac{\sqrt{7}}{2}}$ .

$\frac{\sqrt{\frac{3}{2}}}{\frac{\sqrt{7}}{2}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \sqrt{\frac{3}{2}} (\frac{2}{\sqrt{7}})$

Step 4.2

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\sin (θ) = \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.3

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.2

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.3

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.4

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.6

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

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Step 4.4.6.1

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.6.2

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.6.3

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

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{7}}$

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.

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.4.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{3} \sqrt{2}}{2} \cdot \frac{2}{\sqrt{7}}$

Step 4.5.2

Rewrite the expression.

$\sin (θ) = \sqrt{3} \sqrt{2} (\frac{1}{\sqrt{7}})$

Step 4.6

Combine using the product rule for radicals.

$\sin (θ) = \sqrt{3 \cdot 2} (\frac{1}{\sqrt{7}})$

Step 4.7

Multiply $3$ by $2$ .

$\sin (θ) = \sqrt{6} (\frac{1}{\sqrt{7}})$

Step 4.8

Combine $\sqrt{6}$ and $\frac{1}{\sqrt{7}}$ .

$\sin (θ) = \frac{\sqrt{6}}{\sqrt{7}}$

Step 4.9

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (θ) = \frac{\sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.10

Combine and simplify the denominator.

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Step 4.10.1

Multiply $\frac{\sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.10.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.10.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.10.4

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

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.10.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.10.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.10.6.2

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

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.10.6.3

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

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.10.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{7}$

Step 4.10.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{6} \sqrt{7}}{7}$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Combine using the product rule for radicals.

$\sin (θ) = \frac{\sqrt{6 \cdot 7}}{7}$

Step 4.11.2

Multiply $6$ by $7$ .

$\sin (θ) = \frac{\sqrt{42}}{7}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{42}}{7} \approx 0.92582009$