Trigonometry Examples

$(6, \frac{21 π}{4})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(6, \frac{21 π}{4})$ , draw the triangle between the three points $(0, 0)$ , $(6, 0)$ , and $(6, \frac{21 π}{4})$ .

Opposite : $\frac{21 π}{4}$

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 $2$ .

$\sqrt{36 + {(\frac{21 π}{4})}^{2}}$

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 $\frac{21 π}{4}$ .

$\sqrt{36 + \frac{{(21 π)}^{2}}{4^{2}}}$

Step 2.2.2

Apply the product rule to $21 π$ .

$\sqrt{36 + \frac{21^{2} π^{2}}{4^{2}}}$

Step 2.3

Raise $21$ to the power of $2$ .

$\sqrt{36 + \frac{441 π^{2}}{4^{2}}}$

Step 2.4

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

$\sqrt{36 + \frac{441 π^{2}}{16}}$

Step 2.5

To write $36$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{36 \cdot \frac{16}{16} + \frac{441 π^{2}}{16}}$

Step 2.6

Combine $36$ and $\frac{16}{16}$ .

$\sqrt{\frac{36 \cdot 16}{16} + \frac{441 π^{2}}{16}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Combine the numerators over the common denominator.

$\sqrt{\frac{36 \cdot 16 + 441 π^{2}}{16}}$

Step 2.7.2

Multiply $36$ by $16$ .

$\sqrt{\frac{576 + 441 π^{2}}{16}}$

Step 2.8

Rewrite $\sqrt{\frac{576 + 441 π^{2}}{16}}$ as $\frac{\sqrt{576 + 441 π^{2}}}{\sqrt{16}}$ .

$\frac{\sqrt{576 + 441 π^{2}}}{\sqrt{16}}$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Rewrite $576 + 441 π^{2}$ as $3^{2} (64 + 49 π^{2})$ .

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Step 2.9.1.1

Factor $9$ out of $576$ .

$\frac{\sqrt{9 (64) + 441 π^{2}}}{\sqrt{16}}$

Step 2.9.1.2

Factor $9$ out of $441 π^{2}$ .

$\frac{\sqrt{9 (64) + 9 (49 π^{2})}}{\sqrt{16}}$

Step 2.9.1.3

Factor $9$ out of $9 (64) + 9 (49 π^{2})$ .

$\frac{\sqrt{9 (64 + 49 π^{2})}}{\sqrt{16}}$

Step 2.9.1.4

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

$\frac{\sqrt{3^{2} (64 + 49 π^{2})}}{\sqrt{16}}$

Step 2.9.2

Pull terms out from under the radical.

$\frac{3 \sqrt{64 + 49 π^{2}}}{\sqrt{16}}$

Step 2.10

Simplify the denominator.

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Step 2.10.1

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

$\frac{3 \sqrt{64 + 49 π^{2}}}{\sqrt{4^{2}}}$

Step 2.10.2

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

$\frac{3 \sqrt{64 + 49 π^{2}}}{4}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{21 π}{4}}{\frac{3 \sqrt{64 + 49 π^{2}}}{4}}$ .

$\frac{\frac{21 π}{4}}{\frac{3 \sqrt{64 + 49 π^{2}}}{4}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{21 π}{4} \cdot \frac{4}{3 \sqrt{64 + 49 π^{2}}}$

Step 4.2

Combine.

$\sin (θ) = \frac{21 π \cdot 4}{4 (3 \sqrt{64 + 49 π^{2}})}$

Step 4.3

Cancel the common factor of $21$ and $3$ .

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Step 4.3.1

Factor $3$ out of $21 π \cdot 4$ .

$\sin (θ) = \frac{3 (7 π \cdot 4)}{4 (3 \sqrt{64 + 49 π^{2}})}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $3$ out of $4 (3 \sqrt{64 + 49 π^{2}})$ .

$\sin (θ) = \frac{3 (7 π \cdot 4)}{3 (4 (\sqrt{64 + 49 π^{2}}))}$

Step 4.3.2.2

Cancel the common factor.

$\sin (θ) = \frac{3 (7 π \cdot 4)}{3 (4 (\sqrt{64 + 49 π^{2}}))}$

Step 4.3.2.3

Rewrite the expression.

$\sin (θ) = \frac{7 π \cdot 4}{4 (\sqrt{64 + 49 π^{2}})}$

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Cancel the common factor.

$\sin (θ) = \frac{7 π \cdot 4}{4 \sqrt{64 + 49 π^{2}}}$

Step 4.4.2

Rewrite the expression.

$\sin (θ) = \frac{7 π}{\sqrt{64 + 49 π^{2}}}$

Step 4.5

Multiply $\frac{7 π}{\sqrt{64 + 49 π^{2}}}$ by $\frac{\sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}}}$ .

$\sin (θ) = \frac{7 π}{\sqrt{64 + 49 π^{2}}} \cdot \frac{\sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}}}$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{7 π}{\sqrt{64 + 49 π^{2}}}$ by $\frac{\sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}}}$ .

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}} \sqrt{64 + 49 π^{2}}}$

Step 4.6.2

Raise $\sqrt{64 + 49 π^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}} \sqrt{64 + 49 π^{2}}}$

Step 4.6.3

Raise $\sqrt{64 + 49 π^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{\sqrt{64 + 49 π^{2}} \sqrt{64 + 49 π^{2}}}$

Step 4.6.4

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

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{\sqrt{64 + 49 π^{2}}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{\sqrt{64 + 49 π^{2}}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{64 + 49 π^{2}}}^{2}$ as $64 + 49 π^{2}$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 + 49 π^{2}}$ as ${(64 + 49 π^{2})}^{\frac{1}{2}}$ .

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{({(64 + 49 π^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{(64 + 49 π^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{(64 + 49 π^{2})}^{\frac{2}{2}}}$

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.

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{{(64 + 49 π^{2})}^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{64 + 49 π^{2}}$

Step 4.6.6.5

Simplify.

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{64 + 49 π^{2}}$

Step 5

Approximate the result.

$\sin (θ) = \frac{7 π \sqrt{64 + 49 π^{2}}}{64 + 49 π^{2}} \approx 0.93974924$