Trigonometry Examples

$(6, - 3)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(6, - 3)$ , draw the triangle between the three points $(0, 0)$ , $(6, 0)$ , and $(6, - 3)$ .

Opposite : $- 3$

Adjacent : $6$

Step 2

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

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Step 2.1

Raise $6$ to the power of $2$ .

$\sqrt{36 + {(- 3)}^{2}}$

Step 2.2

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

$\sqrt{36 + 9}$

Step 2.3

Add $36$ and $9$ .

$\sqrt{45}$

Step 2.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 2.4.1

Factor $9$ out of $45$ .

$\sqrt{9 (5)}$

Step 2.4.2

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

$\sqrt{3^{2} \cdot 5}$

Step 2.5

Pull terms out from under the radical.

$3 \sqrt{5}$

Step 3

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

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

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Cancel the common factor of $- 3$ and $3$ .

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Step 4.1.1

Factor $3$ out of $- 3$ .

$\sin (θ) = \frac{3 \cdot - 1}{3 \sqrt{5}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $3$ out of $3 \sqrt{5}$ .

$\sin (θ) = \frac{3 \cdot - 1}{3 (\sqrt{5})}$

Step 4.1.2.2

Cancel the common factor.

$\sin (θ) = \frac{3 \cdot - 1}{3 \sqrt{5}}$

Step 4.1.2.3

Rewrite the expression.

$\sin (θ) = \frac{- 1}{\sqrt{5}}$

Step 4.2

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{\sqrt{5}}$

Step 4.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = - (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.4.4

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

$\sin (θ) = - \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.4.6

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

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Step 4.4.6.1

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

$\sin (θ) = - \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\sin (θ) = - \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\sin (θ) = - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{\sqrt{5}}{5}$

Step 4.4.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{\sqrt{5}}{5}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{\sqrt{5}}{5} \approx - 0.44721359$