Trigonometry Examples

$(- 80, 18)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 80, 18)$ , draw the triangle between the three points $(0, 0)$ , $(- 80, 0)$ , and $(- 80, 18)$ .

Opposite : $18$

Adjacent : $- 80$

Step 2

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

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Step 2.1

Raise $- 80$ to the power of $2$ .

$\sqrt{6400 + {(18)}^{2}}$

Step 2.2

Raise $18$ to the power of $2$ .

$\sqrt{6400 + 324}$

Step 2.3

Add $6400$ and $324$ .

$\sqrt{6724}$

Step 2.4

Rewrite $6724$ as $82^{2}$ .

$\sqrt{82^{2}}$

Step 2.5

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

$82$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 80}{82}$ .

$\frac{- 80}{82}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $- 80$ and $82$ .

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Step 4.1.1

Factor $2$ out of $- 80$ .

$\cos (θ) = \frac{2 (- 40)}{82}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $82$ .

$\cos (θ) = \frac{2 \cdot - 40}{2 \cdot 41}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot - 40}{2 \cdot 41}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{- 40}{41}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{40}{41}$

Step 5

Approximate the result.

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