Trigonometry Examples

$(3, \sqrt{3})$

Step 1

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

Opposite : $\sqrt{3}$

Adjacent : $3$

Step 2

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

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Step 2.1

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(\sqrt{3})}^{2}}$

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

$\sqrt{9 + {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{9 + 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{9 + 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{9 + 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{9 + 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{9 + 3}$

Step 2.3

Add $9$ and $3$ .

$\sqrt{12}$

Step 2.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.4.1

Factor $4$ out of $12$ .

$\sqrt{4 (3)}$

Step 2.4.2

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

$\sqrt{2^{2} \cdot 3}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{3}$

Step 3

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

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

Step 4

Cancel the common factor of $\sqrt{3}$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\sin (θ) = \frac{1}{2}$

Step 5

Approximate the result.

$\sin (θ) = \frac{1}{2} \approx 0.5$