Trigonometry Examples

$(- \frac{40}{41}, - \frac{9}{41})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{40}{41}, - \frac{9}{41})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{40}{41}, 0)$ , and $(- \frac{40}{41}, - \frac{9}{41})$ .

Opposite : $- \frac{9}{41}$

Adjacent : $- \frac{40}{41}$

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 $- \frac{40}{41}$ .

$\sqrt{{(- 1)}^{2} {(\frac{40}{41})}^{2} + {(- \frac{9}{41})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{40}{41}$ .

$\sqrt{{(- 1)}^{2} \frac{40^{2}}{41^{2}} + {(- \frac{9}{41})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{40^{2}}{41^{2}} + {(- \frac{9}{41})}^{2}}$

Step 2.3

Multiply $\frac{40^{2}}{41^{2}}$ by $1$ .

$\sqrt{\frac{40^{2}}{41^{2}} + {(- \frac{9}{41})}^{2}}$

Step 2.4

Raise $40$ to the power of $2$ .

$\sqrt{\frac{1600}{41^{2}} + {(- \frac{9}{41})}^{2}}$

Step 2.5

Raise $41$ to the power of $2$ .

$\sqrt{\frac{1600}{1681} + {(- \frac{9}{41})}^{2}}$

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 $- \frac{9}{41}$ .

$\sqrt{\frac{1600}{1681} + {(- 1)}^{2} {(\frac{9}{41})}^{2}}$

Step 2.6.2

Apply the product rule to $\frac{9}{41}$ .

$\sqrt{\frac{1600}{1681} + {(- 1)}^{2} \frac{9^{2}}{41^{2}}}$

Step 2.7

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

$\sqrt{\frac{1600}{1681} + 1 \frac{9^{2}}{41^{2}}}$

Step 2.8

Multiply $\frac{9^{2}}{41^{2}}$ by $1$ .

$\sqrt{\frac{1600}{1681} + \frac{9^{2}}{41^{2}}}$

Step 2.9

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

$\sqrt{\frac{1600}{1681} + \frac{81}{41^{2}}}$

Step 2.10

Raise $41$ to the power of $2$ .

$\sqrt{\frac{1600}{1681} + \frac{81}{1681}}$

Step 2.11

Combine the numerators over the common denominator.

$\sqrt{\frac{1600 + 81}{1681}}$

Step 2.12

Add $1600$ and $81$ .

$\sqrt{\frac{1681}{1681}}$

Step 2.13

Divide $1681$ by $1681$ .

$\sqrt{1}$

Step 2.14

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{40}{41}}{1}$ .

$\frac{- \frac{40}{41}}{1}$

Step 4

Divide $- \frac{40}{41}$ by $1$ .

$\cos (θ) = - \frac{40}{41}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{40}{41} \approx - 0.\overline{97560}$