Trigonometry Examples

$(1, - \frac{π}{4})$

Step 1

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

Opposite : $- \frac{π}{4}$

Adjacent : $1$

Step 2

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

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Step 2.1

One to any power is one.

$\sqrt{1 + {(- \frac{π}{4})}^{2}}$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

$\sqrt{1 + 1 \frac{π^{2}}{4^{2}}}$

Step 2.4

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

$\sqrt{1 + \frac{π^{2}}{4^{2}}}$

Step 2.5

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

$\sqrt{1 + \frac{π^{2}}{16}}$

Step 2.6

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{16}{16} + \frac{π^{2}}{16}}$

Step 2.7

Combine the numerators over the common denominator.

$\sqrt{\frac{16 + π^{2}}{16}}$

Step 2.8

Rewrite $\sqrt{\frac{16 + π^{2}}{16}}$ as $\frac{\sqrt{16 + π^{2}}}{\sqrt{16}}$ .

$\frac{\sqrt{16 + π^{2}}}{\sqrt{16}}$

Step 2.9

Simplify the denominator.

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Step 2.9.1

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

$\frac{\sqrt{16 + π^{2}}}{\sqrt{4^{2}}}$

Step 2.9.2

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

$\frac{\sqrt{16 + π^{2}}}{4}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{π}{4}}{\frac{\sqrt{16 + π^{2}}}{4}}$ .

$\frac{- \frac{π}{4}}{\frac{\sqrt{16 + π^{2}}}{4}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = - \frac{π}{4} \cdot \frac{4}{\sqrt{16 + π^{2}}}$

Step 4.2

Cancel the common factor of $4$ .

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Step 4.2.1

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

$\sin (θ) = \frac{- π}{4} \cdot \frac{4}{\sqrt{16 + π^{2}}}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{- π}{4} \cdot \frac{4}{\sqrt{16 + π^{2}}}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = - π \frac{1}{\sqrt{16 + π^{2}}}$

Step 4.3

Combine $\frac{1}{\sqrt{16 + π^{2}}}$ and $π$ .

$\sin (θ) = - \frac{π}{\sqrt{16 + π^{2}}}$

Step 4.4

Multiply $\frac{π}{\sqrt{16 + π^{2}}}$ by $\frac{\sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}}}$ .

$\sin (θ) = - (\frac{π}{\sqrt{16 + π^{2}}} \cdot \frac{\sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}}})$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{π}{\sqrt{16 + π^{2}}}$ by $\frac{\sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}}}$ .

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}} \sqrt{16 + π^{2}}}$

Step 4.5.2

Raise $\sqrt{16 + π^{2}}$ to the power of $1$ .

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}} \sqrt{16 + π^{2}}}$

Step 4.5.3

Raise $\sqrt{16 + π^{2}}$ to the power of $1$ .

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{\sqrt{16 + π^{2}} \sqrt{16 + π^{2}}}$

Step 4.5.4

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

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{\sqrt{16 + π^{2}}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{\sqrt{16 + π^{2}}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{16 + π^{2}}}^{2}$ as $16 + π^{2}$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 + π^{2}}$ as ${(16 + π^{2})}^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{({(16 + π^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{(16 + π^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{(16 + π^{2})}^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{{(16 + π^{2})}^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{16 + π^{2}}$

Step 4.5.6.5

Simplify.

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{16 + π^{2}}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{π \sqrt{16 + π^{2}}}{16 + π^{2}} \approx - 0.61766782$