Trigonometry Examples

(- 9, - 5)

$(- 9, - 5)$

Step 1

To find the

sin (θ)

$\sin (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(- 9, - 5)

$(- 9, - 5)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- 9, 0)

$(- 9, 0)$ , and

(- 9, - 5)

$(- 9, - 5)$ .

Opposite :

- 5

$- 5$

Adjacent :

- 9

$- 9$

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise

- 9

$- 9$ to the power of

2

$2$ .

\sqrt{81 + {(- 5)}^{2}}

$\sqrt{81 + {(- 5)}^{2}}$

Step 2.2

Raise

- 5

$- 5$ to the power of

2

$2$ .

\sqrt{81 + 25}

$\sqrt{81 + 25}$

Step 2.3

Add

81

$81$ and

25

$25$ .

\sqrt{106}

$\sqrt{106}$

\sqrt{106}

$\sqrt{106}$

Step 3

sin (θ) = \frac{Opposite}{Hypotenuse}

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore

sin (θ) = \frac{- 5}{\sqrt{106}}

$\sin (θ) = \frac{- 5}{\sqrt{106}}$ .

\frac{- 5}{\sqrt{106}}

$\frac{- 5}{\sqrt{106}}$

Step 4

Simplify

sin (θ)

$\sin (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

sin (θ) = - \frac{5}{\sqrt{106}}

$\sin (θ) = - \frac{5}{\sqrt{106}}$

Step 4.2

Multiply

\frac{5}{\sqrt{106}}

$\frac{5}{\sqrt{106}}$ by

\frac{\sqrt{106}}{\sqrt{106}}

$\frac{\sqrt{106}}{\sqrt{106}}$ .

sin (θ) = - (\frac{5}{\sqrt{106}} \cdot \frac{\sqrt{106}}{\sqrt{106}})

$\sin (θ) = - (\frac{5}{\sqrt{106}} \cdot \frac{\sqrt{106}}{\sqrt{106}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply

\frac{5}{\sqrt{106}}

$\frac{5}{\sqrt{106}}$ by

\frac{\sqrt{106}}{\sqrt{106}}

$\frac{\sqrt{106}}{\sqrt{106}}$ .

sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}

$\sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}$

Step 4.3.2

Raise

\sqrt{106}

$\sqrt{106}$ to the power of

1

$1$ .

sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}

$\sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}$

Step 4.3.3

Raise

\sqrt{106}

$\sqrt{106}$ to the power of

1

$1$ .

sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}

$\sin (θ) = - \frac{5 \sqrt{106}}{\sqrt{106} \sqrt{106}}$

Step 4.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

sin (θ) = - \frac{5 \sqrt{106}}{{\sqrt{106}}^{1 + 1}}

$\sin (θ) = - \frac{5 \sqrt{106}}{{\sqrt{106}}^{1 + 1}}$

Step 4.3.5

Add

1

$1$ and

1

$1$ .

sin (θ) = - \frac{5 \sqrt{106}}{{\sqrt{106}}^{2}}

$\sin (θ) = - \frac{5 \sqrt{106}}{{\sqrt{106}}^{2}}$

Step 4.3.6

Rewrite

{\sqrt{106}}^{2}

${\sqrt{106}}^{2}$ as

106

$106$ .

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Step 4.3.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{106}

$\sqrt{106}$ as

106^{\frac{1}{2}}

$106^{\frac{1}{2}}$ .

sin (θ) = - \frac{5 \sqrt{106}}{{(106^{\frac{1}{2}})}^{2}}

$\sin (θ) = - \frac{5 \sqrt{106}}{{(106^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

sin (θ) = - \frac{5 \sqrt{106}}{106^{\frac{1}{2} \cdot 2}}

$\sin (θ) = - \frac{5 \sqrt{106}}{106^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

sin (θ) = - \frac{5 \sqrt{106}}{106^{\frac{2}{2}}}

$\sin (θ) = - \frac{5 \sqrt{106}}{106^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of

2

$2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{5 \sqrt{106}}{106^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{5 \sqrt{106}}{106}$

Step 4.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{5 \sqrt{106}}{106}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{5 \sqrt{106}}{106} \approx - 0.48564293$