Trigonometry Examples

$(- 3, \frac{- 2 π}{3})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 3, \frac{- 2 π}{3})$ , draw the triangle between the three points $(0, 0)$ , $(- 3, 0)$ , and $(- 3, \frac{- 2 π}{3})$ .

Opposite : $\frac{- 2 π}{3}$

Adjacent : $- 3$

Step 2

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

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Step 2.1

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

$\sqrt{9 + {(\frac{- 2 π}{3})}^{2}}$

Step 2.2

Move the negative in front of the fraction.

$\sqrt{9 + {(- \frac{2 π}{3})}^{2}}$

Step 2.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1

Apply the product rule to $- \frac{2 π}{3}$ .

$\sqrt{9 + {(- 1)}^{2} {(\frac{2 π}{3})}^{2}}$

Step 2.3.2

Apply the product rule to $\frac{2 π}{3}$ .

$\sqrt{9 + {(- 1)}^{2} \frac{{(2 π)}^{2}}{3^{2}}}$

Step 2.3.3

Apply the product rule to $2 π$ .

$\sqrt{9 + {(- 1)}^{2} \frac{2^{2} π^{2}}{3^{2}}}$

Step 2.4

Raise $- 1$ to the power of $2$ .

$\sqrt{9 + 1 \frac{2^{2} π^{2}}{3^{2}}}$

Step 2.5

Multiply $\frac{2^{2} π^{2}}{3^{2}}$ by $1$ .

$\sqrt{9 + \frac{2^{2} π^{2}}{3^{2}}}$

Step 2.6

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

$\sqrt{9 + \frac{4 π^{2}}{3^{2}}}$

Step 2.7

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

$\sqrt{9 + \frac{4 π^{2}}{9}}$

Step 2.8

To write $9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{9 \cdot \frac{9}{9} + \frac{4 π^{2}}{9}}$

Step 2.9

Combine $9$ and $\frac{9}{9}$ .

$\sqrt{\frac{9 \cdot 9}{9} + \frac{4 π^{2}}{9}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Combine the numerators over the common denominator.

$\sqrt{\frac{9 \cdot 9 + 4 π^{2}}{9}}$

Step 2.10.2

Multiply $9$ by $9$ .

$\sqrt{\frac{81 + 4 π^{2}}{9}}$

Step 2.11

Rewrite $\sqrt{\frac{81 + 4 π^{2}}{9}}$ as $\frac{\sqrt{81 + 4 π^{2}}}{\sqrt{9}}$ .

$\frac{\sqrt{81 + 4 π^{2}}}{\sqrt{9}}$

Step 2.12

Simplify the denominator.

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Step 2.12.1

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

$\frac{\sqrt{81 + 4 π^{2}}}{\sqrt{3^{2}}}$

Step 2.12.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{81 + 4 π^{2}}}{3}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 3}{\frac{\sqrt{81 + 4 π^{2}}}{3}}$ .

$\frac{- 3}{\frac{\sqrt{81 + 4 π^{2}}}{3}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - 3 \frac{3}{\sqrt{81 + 4 π^{2}}}$

Step 4.2

Multiply $\frac{3}{\sqrt{81 + 4 π^{2}}}$ by $\frac{\sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}}}$ .

$\cos (θ) = - 3 (\frac{3}{\sqrt{81 + 4 π^{2}}} \cdot \frac{\sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{3}{\sqrt{81 + 4 π^{2}}}$ by $\frac{\sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}}}$ .

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}} \sqrt{81 + 4 π^{2}}}$

Step 4.3.2

Raise $\sqrt{81 + 4 π^{2}}$ to the power of $1$ .

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}} \sqrt{81 + 4 π^{2}}}$

Step 4.3.3

Raise $\sqrt{81 + 4 π^{2}}$ to the power of $1$ .

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{\sqrt{81 + 4 π^{2}} \sqrt{81 + 4 π^{2}}}$

Step 4.3.4

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

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{\sqrt{81 + 4 π^{2}}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{\sqrt{81 + 4 π^{2}}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{81 + 4 π^{2}}}^{2}$ as $81 + 4 π^{2}$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{81 + 4 π^{2}}$ as ${(81 + 4 π^{2})}^{\frac{1}{2}}$ .

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{({(81 + 4 π^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{(81 + 4 π^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{(81 + 4 π^{2})}^{\frac{2}{2}}}$

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.

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{{(81 + 4 π^{2})}^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$

Step 4.3.6.5

Simplify.

$\cos (θ) = - 3 \frac{3 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$

Step 4.4

Multiply $- 3 \frac{3 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$ .

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Step 4.4.1

Combine $- 3$ and $\frac{3 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$ .

$\cos (θ) = \frac{- 3 (3 \sqrt{81 + 4 π^{2}})}{81 + 4 π^{2}}$

Step 4.4.2

Multiply $3$ by $- 3$ .

$\cos (θ) = \frac{- 9 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$

Step 4.5

Move the negative in front of the fraction.

$\cos (θ) = - \frac{9 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{9 \sqrt{81 + 4 π^{2}}}{81 + 4 π^{2}} \approx - 0.81995096$