Trigonometry Examples

$(- 2, - 9)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, - 9)$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, - 9)$ .

Opposite : $- 9$

Adjacent : $- 2$

Step 2

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

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Step 2.1

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

$\sqrt{4 + {(- 9)}^{2}}$

Step 2.2

Raise $- 9$ to the power of $2$ .

$\sqrt{4 + 81}$

Step 2.3

Add $4$ and $81$ .

$\sqrt{85}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 9}{\sqrt{85}}$ .

$\frac{- 9}{\sqrt{85}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{9}{\sqrt{85}}$

Step 4.2

Multiply $\frac{9}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\sin (θ) = - (\frac{9}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{9}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\sin (θ) = - \frac{9 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.2

Raise $\sqrt{85}$ to the power of $1$ .

$\sin (θ) = - \frac{9 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.3

Raise $\sqrt{85}$ to the power of $1$ .

$\sin (θ) = - \frac{9 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.4

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

$\sin (θ) = - \frac{9 \sqrt{85}}{{\sqrt{85}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{9 \sqrt{85}}{{\sqrt{85}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{85}}^{2}$ as $85$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{85}$ as $85^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{9 \sqrt{85}}{{(85^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\sin (θ) = - \frac{9 \sqrt{85}}{85^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\sin (θ) = - \frac{9 \sqrt{85}}{85^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{9 \sqrt{85}}{85^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{9 \sqrt{85}}{85}$

Step 4.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{9 \sqrt{85}}{85}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{9 \sqrt{85}}{85} \approx - 0.97618706$