Trigonometry Examples

$(1, \sqrt{2})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(1, \sqrt{2})$ , draw the triangle between the three points $(0, 0)$ , $(1, 0)$ , and $(1, \sqrt{2})$ .

Opposite : $\sqrt{2}$

Adjacent : $1$

Step 2

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

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Step 2.1

One to any power is one.

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

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Evaluate the exponent.

$\sqrt{1 + 2}$

Step 2.3

Add $1$ and $2$ .

$\sqrt{3}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{2}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Add $1$ and $1$ .

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

Step 4.2.6

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

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Step 4.2.6.1

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

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

Step 4.2.6.2

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

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

Step 4.2.6.3

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

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

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

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

Step 4.2.6.4.2

Rewrite the expression.

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

Step 4.2.6.5

Evaluate the exponent.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine using the product rule for radicals.

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

Step 4.3.2

Multiply $2$ by $3$ .

$\sin (θ) = \frac{\sqrt{6}}{3}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{6}}{3} \approx 0.81649658$