Trigonometry Examples

$(- 2, - \sqrt{5})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, - \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, - \sqrt{5})$ .

Opposite : $- \sqrt{5}$

Adjacent : $- 2$

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

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

$\sqrt{4 + {(- \sqrt{5})}^{2}}$

Step 2.1.2

Apply the product rule to $- \sqrt{5}$ .

$\sqrt{4 + {(- 1)}^{2} {\sqrt{5}}^{2}}$

Step 2.1.3

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

$\sqrt{4 + 1 {\sqrt{5}}^{2}}$

Step 2.1.4

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$\sqrt{4 + {\sqrt{5}}^{2}}$

Step 2.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{4 + {(5^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{4 + 5^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{4 + 5^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{4 + 5^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{4 + 5^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{4 + 5}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Add $4$ and $5$ .

$\sqrt{9}$

Step 2.3.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 2.4

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

$3$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \sqrt{5}}{3}$ .

$\frac{- \sqrt{5}}{3}$

Step 4

Move the negative in front of the fraction.

$\sin (θ) = - \frac{\sqrt{5}}{3}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{\sqrt{5}}{3} \approx - 0.74535599$