Trigonometry Examples

(- 2, 9)

$(- 2, 9)$

Step 1

To find the

cos (θ)

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

(0, 0)

$(0, 0)$ and

(- 2, 9)

$(- 2, 9)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- 2, 0)

$(- 2, 0)$ , and

(- 2, 9)

$(- 2, 9)$ .

Opposite :

9

$9$

Adjacent :

- 2

$- 2$

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

- 2

$- 2$ to the power of

2

$2$ .

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

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

Step 2.2

Raise

9

$9$ to the power of

2

$2$ .

\sqrt{4 + 81}

$\sqrt{4 + 81}$

Step 2.3

Add

4

$4$ and

81

$81$ .

\sqrt{85}

$\sqrt{85}$

\sqrt{85}

$\sqrt{85}$

Step 3

cos (θ) = \frac{Adjacent}{Hypotenuse}

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore

cos (θ) = \frac{- 2}{\sqrt{85}}

$\cos (θ) = \frac{- 2}{\sqrt{85}}$ .

\frac{- 2}{\sqrt{85}}

$\frac{- 2}{\sqrt{85}}$

Step 4

Simplify

cos (θ)

$\cos (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

cos (θ) = - \frac{2}{\sqrt{85}}

$\cos (θ) = - \frac{2}{\sqrt{85}}$

Step 4.2

Multiply

\frac{2}{\sqrt{85}}

$\frac{2}{\sqrt{85}}$ by

\frac{\sqrt{85}}{\sqrt{85}}

$\frac{\sqrt{85}}{\sqrt{85}}$ .

$\cos (θ) = - (\frac{2}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply

$\frac{2}{\sqrt{85}}$ by

$\frac{\sqrt{85}}{\sqrt{85}}$ .

$\cos (θ) = - \frac{2 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.2

Raise

$\sqrt{85}$ to the power of

$1$ .

$\cos (θ) = - \frac{2 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.3

Raise

$\sqrt{85}$ to the power of

$1$ .

$\cos (θ) = - \frac{2 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.3.4

Use the power rule

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

$\cos (θ) = - \frac{2 \sqrt{85}}{{\sqrt{85}}^{1 + 1}}$

Step 4.3.5

Add

$1$ and

$1$ .

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

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

$85^{\frac{1}{2}}$ .

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

Step 4.3.6.2

Apply the power rule and multiply exponents,

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

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

Step 4.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

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.

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

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{2 \sqrt{85}}{85}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{2 \sqrt{85}}{85}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{2 \sqrt{85}}{85} \approx - 0.21693045$

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