Trigonometry Examples

$(2, 11)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(2, 11)$ , draw the triangle between the three points $(0, 0)$ , $(2, 0)$ , and $(2, 11)$ .

Opposite : $11$

Adjacent : $2$

Step 2

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

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Step 2.1

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

$\sqrt{4 + {(11)}^{2}}$

Step 2.2

Raise $11$ to the power of $2$ .

$\sqrt{4 + 121}$

Step 2.3

Add $4$ and $121$ .

$\sqrt{125}$

Step 2.4

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 2.4.1

Factor $25$ out of $125$ .

$\sqrt{25 (5)}$

Step 2.4.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 5}$

Step 2.5

Pull terms out from under the radical.

$5 \sqrt{5}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{11}{5 \sqrt{5}}$ .

$\frac{11}{5 \sqrt{5}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{11}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = \frac{11}{5 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{11}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 \sqrt{5} \sqrt{5}}$

Step 4.2.2

Move $\sqrt{5}$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 (\sqrt{5} \sqrt{5})}$

Step 4.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 (\sqrt{5} \sqrt{5})}$

Step 4.2.4

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 (\sqrt{5} \sqrt{5})}$

Step 4.2.5

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

$\sin (θ) = \frac{11 \sqrt{5}}{5 {\sqrt{5}}^{1 + 1}}$

Step 4.2.6

Add $1$ and $1$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 {\sqrt{5}}^{2}}$

Step 4.2.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (θ) = \frac{11 \sqrt{5}}{5 {(5^{\frac{1}{2}})}^{2}}$

Step 4.2.7.2

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

$\sin (θ) = \frac{11 \sqrt{5}}{5 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 4.2.7.3

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

$\sin (θ) = \frac{11 \sqrt{5}}{5 \cdot 5^{\frac{2}{2}}}$

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.

$\sin (θ) = \frac{11 \sqrt{5}}{5 \cdot 5^{\frac{2}{2}}}$

Step 4.2.7.4.2

Rewrite the expression.

$\sin (θ) = \frac{11 \sqrt{5}}{5 \cdot 5}$

Step 4.2.7.5

Evaluate the exponent.

$\sin (θ) = \frac{11 \sqrt{5}}{5 \cdot 5}$

Step 4.3

Multiply $5$ by $5$ .

$\sin (θ) = \frac{11 \sqrt{5}}{25}$

Step 5

Approximate the result.

$\sin (θ) = \frac{11 \sqrt{5}}{25} \approx 0.98386991$