Trigonometry Examples

$(- \frac{1}{4}, - \frac{3}{4})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{4}, - \frac{3}{4})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{4}, 0)$ , and $(- \frac{1}{4}, - \frac{3}{4})$ .

Opposite : $- \frac{3}{4}$

Adjacent : $- \frac{1}{4}$

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 $- \frac{1}{4}$ .

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

Step 2.1.2

Apply the product rule to $\frac{1}{4}$ .

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

Step 2.2

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

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

Step 2.3

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

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

Step 2.4

One to any power is one.

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

Step 2.5

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

$\sqrt{\frac{1}{16} + {(- \frac{3}{4})}^{2}}$

Step 2.6

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

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Step 2.6.1

Apply the product rule to $- \frac{3}{4}$ .

$\sqrt{\frac{1}{16} + {(- 1)}^{2} {(\frac{3}{4})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{3}{4}$ .

$\sqrt{\frac{1}{16} + {(- 1)}^{2} \frac{3^{2}}{4^{2}}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

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

$\sqrt{\frac{1}{16} + 1 \frac{3^{2}}{4^{2}}}$

Step 2.7.2

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$\sqrt{\frac{1}{16} + \frac{3^{2}}{4^{2}}}$

Step 2.7.3

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

$\sqrt{\frac{1}{16} + \frac{9}{4^{2}}}$

Step 2.7.4

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

$\sqrt{\frac{1}{16} + \frac{9}{16}}$

Step 2.7.5

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 9}{16}}$

Step 2.7.6

Add $1$ and $9$ .

$\sqrt{\frac{10}{16}}$

Step 2.8

Cancel the common factor of $10$ and $16$ .

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Step 2.8.1

Factor $2$ out of $10$ .

$\sqrt{\frac{2 (5)}{16}}$

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $16$ .

$\sqrt{\frac{2 \cdot 5}{2 \cdot 8}}$

Step 2.8.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 5}{2 \cdot 8}}$

Step 2.8.2.3

Rewrite the expression.

$\sqrt{\frac{5}{8}}$

Step 2.9

Rewrite $\sqrt{\frac{5}{8}}$ as $\frac{\sqrt{5}}{\sqrt{8}}$ .

$\frac{\sqrt{5}}{\sqrt{8}}$

Step 2.10

Simplify the denominator.

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Step 2.10.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.10.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{5}}{\sqrt{4 (2)}}$

Step 2.10.1.2

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

$\frac{\sqrt{5}}{\sqrt{2^{2} \cdot 2}}$

Step 2.10.2

Pull terms out from under the radical.

$\frac{\sqrt{5}}{2 \sqrt{2}}$

Step 2.11

Multiply $\frac{\sqrt{5}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{5}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.12

Combine and simplify the denominator.

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Step 2.12.1

Multiply $\frac{\sqrt{5}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{5} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 2.12.2

Move $\sqrt{2}$ .

$\frac{\sqrt{5} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 2.12.3

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

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

Step 2.12.4

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

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

Step 2.12.5

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

$\frac{\sqrt{5} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 2.12.6

Add $1$ and $1$ .

$\frac{\sqrt{5} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 2.12.7

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

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Step 2.12.7.1

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

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

Step 2.12.7.2

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

$\frac{\sqrt{5} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.12.7.3

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

$\frac{\sqrt{5} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.12.7.4

Cancel the common factor of $2$ .

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Step 2.12.7.4.1

Cancel the common factor.

$\frac{\sqrt{5} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.12.7.4.2

Rewrite the expression.

$\frac{\sqrt{5} \sqrt{2}}{2 \cdot 2^{1}}$

Step 2.12.7.5

Evaluate the exponent.

$\frac{\sqrt{5} \sqrt{2}}{2 \cdot 2}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 2}}{2 \cdot 2}$

Step 2.13.2

Multiply $5$ by $2$ .

$\frac{\sqrt{10}}{2 \cdot 2}$

Step 2.14

Multiply $2$ by $2$ .

$\frac{\sqrt{10}}{4}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{3}{4}}{\frac{\sqrt{10}}{4}}$ .

$\frac{- \frac{3}{4}}{\frac{\sqrt{10}}{4}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = - \frac{3}{4} \cdot \frac{4}{\sqrt{10}}$

Step 4.2

Cancel the common factor of $4$ .

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Step 4.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\sin (θ) = \frac{- 3}{4} \cdot \frac{4}{\sqrt{10}}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{- 3}{4} \cdot \frac{4}{\sqrt{10}}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = - 3 \frac{1}{\sqrt{10}}$

Step 4.3

Combine $- 3$ and $\frac{1}{\sqrt{10}}$ .

$\sin (θ) = \frac{- 3}{\sqrt{10}}$

Step 4.4

Move the negative in front of the fraction.

$\sin (θ) = - \frac{3}{\sqrt{10}}$

Step 4.5

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.6.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.6.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.6.4

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

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

Step 4.6.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{3 \sqrt{10}}{10} \approx - 0.94868329$