Trigonometry Examples

$(2, \sqrt{2})$

Step 1

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

Opposite : $\sqrt{2}$

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 $2$ .

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

Step 2.2

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

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Step 2.2.1

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

$\sqrt{4 + {(2^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{4 + 2^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{4 + 2^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{4 + 2^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{4 + 2^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{4 + 2}$

Step 2.3

Add $4$ and $2$ .

$\sqrt{6}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{2}}{\sqrt{6}}$ .

$\frac{\sqrt{2}}{\sqrt{6}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

$\sin (θ) = \sqrt{\frac{2}{6}}$

Step 4.2

Cancel the common factor of $2$ and $6$ .

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Step 4.2.1

Factor $2$ out of $2$ .

$\sin (θ) = \sqrt{\frac{2 (1)}{6}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $6$ .

$\sin (θ) = \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 4.2.2.2

Cancel the common factor.

$\sin (θ) = \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 4.2.2.3

Rewrite the expression.

$\sin (θ) = \sqrt{\frac{1}{3}}$

Step 4.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\sin (θ) = \frac{\sqrt{1}}{\sqrt{3}}$

Step 4.4

Any root of $1$ is $1$ .

$\sin (θ) = \frac{1}{\sqrt{3}}$

Step 4.5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (θ) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.4

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

$\sin (θ) = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.6.6

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

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Step 4.6.6.1

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

$\sin (θ) = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

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.

$\sin (θ) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{3}}{3}$

Step 4.6.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{3}}{3}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{3}}{3} \approx 0.57735026$