Trigonometry Examples

$(2 \sqrt{3}, - 2)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(2 \sqrt{3}, - 2)$ , draw the triangle between the three points $(0, 0)$ , $(2 \sqrt{3}, 0)$ , and $(2 \sqrt{3}, - 2)$ .

Opposite : $- 2$

Adjacent : $2 \sqrt{3}$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $2 \sqrt{3}$ .

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

Step 2.1.2

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

$\sqrt{4 {\sqrt{3}}^{2} + {(- 2)}^{2}}$

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

$\sqrt{4 {(3^{\frac{1}{2}})}^{2} + {(- 2)}^{2}}$

Step 2.2.2

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

$\sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} + {(- 2)}^{2}}$

Step 2.2.3

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

$\sqrt{4 \cdot 3^{\frac{2}{2}} + {(- 2)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{4 \cdot 3^{\frac{2}{2}} + {(- 2)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 3^{1} + {(- 2)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{4 \cdot 3 + {(- 2)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $4$ by $3$ .

$\sqrt{12 + {(- 2)}^{2}}$

Step 2.3.2

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

$\sqrt{12 + 4}$

Step 2.3.3

Add $12$ and $4$ .

$\sqrt{16}$

Step 2.3.4

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$4$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 2}{4}$ .

$\frac{- 2}{4}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Cancel the common factor of $- 2$ and $4$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$\sin (θ) = \frac{2 (- 1)}{4}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $4$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

$\sin (θ) = \frac{- 1}{2}$

Step 4.2

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{2}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{1}{2} \approx - 0.5$