Trigonometry Examples

$(- 6, \frac{3 π}{2})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 6, \frac{3 π}{2})$ , draw the triangle between the three points $(0, 0)$ , $(- 6, 0)$ , and $(- 6, \frac{3 π}{2})$ .

Opposite : $\frac{3 π}{2}$

Adjacent : $- 6$

Step 2

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

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Step 2.1

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

$\sqrt{36 + {(\frac{3 π}{2})}^{2}}$

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 $\frac{3 π}{2}$ .

$\sqrt{36 + \frac{{(3 π)}^{2}}{2^{2}}}$

Step 2.2.2

Apply the product rule to $3 π$ .

$\sqrt{36 + \frac{3^{2} π^{2}}{2^{2}}}$

Step 2.3

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

$\sqrt{36 + \frac{9 π^{2}}{2^{2}}}$

Step 2.4

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

$\sqrt{36 + \frac{9 π^{2}}{4}}$

Step 2.5

To write $36$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{36 \cdot \frac{4}{4} + \frac{9 π^{2}}{4}}$

Step 2.6

Combine $36$ and $\frac{4}{4}$ .

$\sqrt{\frac{36 \cdot 4}{4} + \frac{9 π^{2}}{4}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Combine the numerators over the common denominator.

$\sqrt{\frac{36 \cdot 4 + 9 π^{2}}{4}}$

Step 2.7.2

Multiply $36$ by $4$ .

$\sqrt{\frac{144 + 9 π^{2}}{4}}$

Step 2.8

Rewrite $\sqrt{\frac{144 + 9 π^{2}}{4}}$ as $\frac{\sqrt{144 + 9 π^{2}}}{\sqrt{4}}$ .

$\frac{\sqrt{144 + 9 π^{2}}}{\sqrt{4}}$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Rewrite $144 + 9 π^{2}$ as $3^{2} (16 + π^{2})$ .

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Step 2.9.1.1

Factor $9$ out of $144$ .

$\frac{\sqrt{9 (16) + 9 π^{2}}}{\sqrt{4}}$

Step 2.9.1.2

Factor $9$ out of $9 π^{2}$ .

$\frac{\sqrt{9 (16) + 9 (π^{2})}}{\sqrt{4}}$

Step 2.9.1.3

Factor $9$ out of $9 (16) + 9 (π^{2})$ .

$\frac{\sqrt{9 (16 + π^{2})}}{\sqrt{4}}$

Step 2.9.1.4

Rewrite $9$ as $3^{2}$ .

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

Step 2.9.2

Pull terms out from under the radical.

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

Step 2.10

Simplify the denominator.

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Step 2.10.1

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

$\frac{3 \sqrt{16 + π^{2}}}{\sqrt{2^{2}}}$

Step 2.10.2

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

$\frac{3 \sqrt{16 + π^{2}}}{2}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{3 π}{2}}{\frac{3 \sqrt{16 + π^{2}}}{2}}$ .

$\frac{\frac{3 π}{2}}{\frac{3 \sqrt{16 + π^{2}}}{2}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{3 π}{2} \cdot \frac{2}{3 \sqrt{16 + π^{2}}}$

Step 4.2

Combine.

$\sin (θ) = \frac{3 π \cdot 2}{2 (3 \sqrt{16 + π^{2}})}$

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

$\sin (θ) = \frac{3 π \cdot 2}{2 (3 \sqrt{16 + π^{2}})}$

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

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 4.6

Combine and simplify the denominator.

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Step 4.6.1

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 4.6.2

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 4.6.3

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 4.6.4

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 4.6.5

Add $1$ and $1$ .

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

Step 4.6.6

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

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Step 4.6.6.1

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 4.6.6.2

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 4.6.6.3

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

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

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.

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

Step 4.6.6.4.2

Rewrite the expression.

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

Step 4.6.6.5

Simplify.

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

Step 5

Approximate the result.

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