Trigonometry Examples

$(\sqrt{3}, - 1)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\sqrt{3}, - 1)$ , draw the triangle between the three points $(0, 0)$ , $(\sqrt{3}, 0)$ , and $(\sqrt{3}, - 1)$ .

Opposite : $- 1$

Adjacent : $\sqrt{3}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Evaluate the exponent.

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Raise $- 1$ to the power of $2$ .

$\sqrt{3 + 1}$

Step 2.2.2

Add $3$ and $1$ .

$\sqrt{4}$

Step 2.2.3

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

$\sqrt{2^{2}}$

Step 2.3

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

$2$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 1}{2}$ .

$\frac{- 1}{2}$

Step 4

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{2}$

Step 5

Approximate the result.

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