Trigonometry Examples

$(\sqrt{3}, 5)$

Step 1

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

Opposite : $5$

Adjacent : $\sqrt{3}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Evaluate the exponent.

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

Step 2.2

Simplify the expression.

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Step 2.2.1

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

$\sqrt{3 + 25}$

Step 2.2.2

Add $3$ and $25$ .

$\sqrt{28}$

Step 2.3

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.3.1

Factor $4$ out of $28$ .

$\sqrt{4 (7)}$

Step 2.3.2

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

$\sqrt{2^{2} \cdot 7}$

Step 2.4

Pull terms out from under the radical.

$2 \sqrt{7}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{5}{2 \sqrt{7}}$ .

$\frac{5}{2 \sqrt{7}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{5}{2 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (θ) = \frac{5}{2 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{5}{2 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 \sqrt{7} \sqrt{7}}$

Step 4.2.2

Move $\sqrt{7}$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 (\sqrt{7} \sqrt{7})}$

Step 4.2.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 (\sqrt{7} \sqrt{7})}$

Step 4.2.4

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 (\sqrt{7} \sqrt{7})}$

Step 4.2.5

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

$\sin (θ) = \frac{5 \sqrt{7}}{2 {\sqrt{7}}^{1 + 1}}$

Step 4.2.6

Add $1$ and $1$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 {\sqrt{7}}^{2}}$

Step 4.2.7

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sin (θ) = \frac{5 \sqrt{7}}{2 {(7^{\frac{1}{2}})}^{2}}$

Step 4.2.7.2

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

$\sin (θ) = \frac{5 \sqrt{7}}{2 \cdot 7^{\frac{1}{2} \cdot 2}}$

Step 4.2.7.3

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

$\sin (θ) = \frac{5 \sqrt{7}}{2 \cdot 7^{\frac{2}{2}}}$

Step 4.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.7.4.1

Cancel the common factor.

$\sin (θ) = \frac{5 \sqrt{7}}{2 \cdot 7^{\frac{2}{2}}}$

Step 4.2.7.4.2

Rewrite the expression.

$\sin (θ) = \frac{5 \sqrt{7}}{2 \cdot 7}$

Step 4.2.7.5

Evaluate the exponent.

$\sin (θ) = \frac{5 \sqrt{7}}{2 \cdot 7}$

Step 4.3

Multiply $2$ by $7$ .

$\sin (θ) = \frac{5 \sqrt{7}}{14}$

Step 5

Approximate the result.

$\sin (θ) = \frac{5 \sqrt{7}}{14} \approx 0.94491118$