Trigonometry Examples

$(16 \sqrt{3}, 16)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(16 \sqrt{3}, 16)$ , draw the triangle between the three points $(0, 0)$ , $(16 \sqrt{3}, 0)$ , and $(16 \sqrt{3}, 16)$ .

Opposite : $16$

Adjacent : $16 \sqrt{3}$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $16 \sqrt{3}$ .

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

Step 2.1.2

Raise $16$ to the power of $2$ .

$\sqrt{256 {\sqrt{3}}^{2} + {(16)}^{2}}$

Step 2.2

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

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{256 {(3^{\frac{1}{2}})}^{2} + {(16)}^{2}}$

Step 2.2.2

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

$\sqrt{256 \cdot 3^{\frac{1}{2} \cdot 2} + {(16)}^{2}}$

Step 2.2.3

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

$\sqrt{256 \cdot 3^{\frac{2}{2}} + {(16)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{256 \cdot 3^{\frac{2}{2}} + {(16)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{256 \cdot 3^{1} + {(16)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{256 \cdot 3 + {(16)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $256$ by $3$ .

$\sqrt{768 + {(16)}^{2}}$

Step 2.3.2

Raise $16$ to the power of $2$ .

$\sqrt{768 + 256}$

Step 2.3.3

Add $768$ and $256$ .

$\sqrt{1024}$

Step 2.3.4

Rewrite $1024$ as $32^{2}$ .

$\sqrt{32^{2}}$

Step 2.3.5

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

$32$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{16}{32}$ .

$\frac{16}{32}$

Step 4

Cancel the common factor of $16$ and $32$ .

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Step 4.1

Factor $16$ out of $16$ .

$\sin (θ) = \frac{16 (1)}{32}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $16$ out of $32$ .

$\sin (θ) = \frac{16 \cdot 1}{16 \cdot 2}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{16 \cdot 1}{16 \cdot 2}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = \frac{1}{2}$

Step 5

Approximate the result.

$\sin (θ) = \frac{1}{2} \approx 0.5$