Trigonometry Examples

$(5 \sqrt{3}, 5)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(5 \sqrt{3}, 5)$ , draw the triangle between the three points $(0, 0)$ , $(5 \sqrt{3}, 0)$ , and $(5 \sqrt{3}, 5)$ .

Opposite : $5$

Adjacent : $5 \sqrt{3}$

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 $5 \sqrt{3}$ .

$\sqrt{5^{2} {\sqrt{3}}^{2} + {(5)}^{2}}$

Step 2.1.2

Raise $5$ to the power of $2$ .

$\sqrt{25 {\sqrt{3}}^{2} + {(5)}^{2}}$

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 $3^{\frac{1}{2}}$ .

$\sqrt{25 {(3^{\frac{1}{2}})}^{2} + {(5)}^{2}}$

Step 2.2.2

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

$\sqrt{25 \cdot 3^{\frac{1}{2} \cdot 2} + {(5)}^{2}}$

Step 2.2.3

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

$\sqrt{25 \cdot 3^{\frac{2}{2}} + {(5)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{25 \cdot 3^{\frac{2}{2}} + {(5)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{25 \cdot 3^{1} + {(5)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{25 \cdot 3 + {(5)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $25$ by $3$ .

$\sqrt{75 + {(5)}^{2}}$

Step 2.3.2

Raise $5$ to the power of $2$ .

$\sqrt{75 + 25}$

Step 2.3.3

Add $75$ and $25$ .

$\sqrt{100}$

Step 2.3.4

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}}$

Step 2.3.5

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

$10$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{5}{10}$ .

$\frac{5}{10}$

Step 4

Cancel the common factor of $5$ and $10$ .

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Step 4.1

Factor $5$ out of $5$ .

$\sin (θ) = \frac{5 (1)}{10}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $10$ .

$\sin (θ) = \frac{5 \cdot 1}{5 \cdot 2}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{5 \cdot 1}{5 \cdot 2}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = \frac{1}{2}$

Step 5

Approximate the result.

$\sin (θ) = \frac{1}{2} \approx 0.5$