Trigonometry Examples

$(- \frac{1}{3}, - \frac{1}{4})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{3}, - \frac{1}{4})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{3}, 0)$ , and $(- \frac{1}{3}, - \frac{1}{4})$ .

Opposite : $- \frac{1}{4}$

Adjacent : $- \frac{1}{3}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Apply the product rule to $- \frac{1}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{1}{3})}^{2} + {(- \frac{1}{4})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{1^{2}}{3^{2}} + {(- \frac{1}{4})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{1^{2}}{3^{2}} + {(- \frac{1}{4})}^{2}}$

Step 2.3

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

$\sqrt{\frac{1^{2}}{3^{2}} + {(- \frac{1}{4})}^{2}}$

Step 2.4

One to any power is one.

$\sqrt{\frac{1}{3^{2}} + {(- \frac{1}{4})}^{2}}$

Step 2.5

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

$\sqrt{\frac{1}{9} + {(- \frac{1}{4})}^{2}}$

Step 2.6

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

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Step 2.6.1

Apply the product rule to $- \frac{1}{4}$ .

$\sqrt{\frac{1}{9} + {(- 1)}^{2} {(\frac{1}{4})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{1}{4}$ .

$\sqrt{\frac{1}{9} + {(- 1)}^{2} \frac{1^{2}}{4^{2}}}$

Step 2.7

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

$\sqrt{\frac{1}{9} + 1 \frac{1^{2}}{4^{2}}}$

Step 2.8

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$\sqrt{\frac{1}{9} + \frac{1^{2}}{4^{2}}}$

Step 2.9

One to any power is one.

$\sqrt{\frac{1}{9} + \frac{1}{4^{2}}}$

Step 2.10

Raise $4$ to the power of $2$ .

$\sqrt{\frac{1}{9} + \frac{1}{16}}$

Step 2.11

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{\frac{1}{9} \cdot \frac{16}{16} + \frac{1}{16}}$

Step 2.12

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{9} \cdot \frac{16}{16} + \frac{1}{16} \cdot \frac{9}{9}}$

Step 2.13

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.13.1

Multiply $\frac{1}{9}$ by $\frac{16}{16}$ .

$\sqrt{\frac{16}{9 \cdot 16} + \frac{1}{16} \cdot \frac{9}{9}}$

Step 2.13.2

Multiply $9$ by $16$ .

$\sqrt{\frac{16}{144} + \frac{1}{16} \cdot \frac{9}{9}}$

Step 2.13.3

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

$\sqrt{\frac{16}{144} + \frac{9}{16 \cdot 9}}$

Step 2.13.4

Multiply $16$ by $9$ .

$\sqrt{\frac{16}{144} + \frac{9}{144}}$

Step 2.14

Combine the numerators over the common denominator.

$\sqrt{\frac{16 + 9}{144}}$

Step 2.15

Add $16$ and $9$ .

$\sqrt{\frac{25}{144}}$

Step 2.16

Rewrite $\sqrt{\frac{25}{144}}$ as $\frac{\sqrt{25}}{\sqrt{144}}$ .

$\frac{\sqrt{25}}{\sqrt{144}}$

Step 2.17

Simplify the numerator.

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Step 2.17.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{144}}$

Step 2.17.2

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

$\frac{5}{\sqrt{144}}$

Step 2.18

Simplify the denominator.

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Step 2.18.1

Rewrite $144$ as $12^{2}$ .

$\frac{5}{\sqrt{12^{2}}}$

Step 2.18.2

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

$\frac{5}{12}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{1}{3}}{\frac{5}{12}}$ .

$\frac{- \frac{1}{3}}{\frac{5}{12}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - \frac{1}{3} \cdot \frac{12}{5}$

Step 4.2

Cancel the common factor of $3$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\cos (θ) = \frac{- 1}{3} \cdot \frac{12}{5}$

Step 4.2.2

Factor $3$ out of $12$ .

$\cos (θ) = \frac{- 1}{3} \cdot \frac{3 (4)}{5}$

Step 4.2.3

Cancel the common factor.

$\cos (θ) = \frac{- 1}{3} \cdot \frac{3 \cdot 4}{5}$

Step 4.2.4

Rewrite the expression.

$\cos (θ) = - \frac{4}{5}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{4}{5} \approx - 0.8$