Trigonometry Examples

$(9, - 9)$

Step 1

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

Opposite : $- 9$

Adjacent : $9$

Step 2

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

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Step 2.1

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

$\sqrt{81 + {(- 9)}^{2}}$

Step 2.2

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

$\sqrt{81 + 81}$

Step 2.3

Add $81$ and $81$ .

$\sqrt{162}$

Step 2.4

Rewrite $162$ as $9^{2} \cdot 2$ .

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Step 2.4.1

Factor $81$ out of $162$ .

$\sqrt{81 (2)}$

Step 2.4.2

Rewrite $81$ as $9^{2}$ .

$\sqrt{9^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

$9 \sqrt{2}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- 9}{9 \sqrt{2}}$ .

$\frac{- 9}{9 \sqrt{2}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Cancel the common factor of $- 9$ and $9$ .

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Step 4.1.1

Factor $9$ out of $- 9$ .

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $9$ out of $9 \sqrt{2}$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

$\sin (θ) = \frac{- 1}{\sqrt{2}}$

Step 4.2

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{\sqrt{2}}$

Step 4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

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

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Step 4.4.6.1

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

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

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.

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

Step 4.4.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Step 4.4.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{\sqrt{2}}{2} \approx - 0.70710678$