Trigonometry Examples

$(- 8, 10)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 8, 10)$ , draw the triangle between the three points $(0, 0)$ , $(- 8, 0)$ , and $(- 8, 10)$ .

Opposite : $10$

Adjacent : $- 8$

Step 2

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

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Step 2.1

Raise $- 8$ to the power of $2$ .

$\sqrt{64 + {(10)}^{2}}$

Step 2.2

Raise $10$ to the power of $2$ .

$\sqrt{64 + 100}$

Step 2.3

Add $64$ and $100$ .

$\sqrt{164}$

Step 2.4

Rewrite $164$ as $2^{2} \cdot 41$ .

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Step 2.4.1

Factor $4$ out of $164$ .

$\sqrt{4 (41)}$

Step 2.4.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 41}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{41}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 8}{2 \sqrt{41}}$ .

$\frac{- 8}{2 \sqrt{41}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $- 8$ and $2$ .

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Step 4.1.1

Factor $2$ out of $- 8$ .

$\cos (θ) = \frac{2 \cdot - 4}{2 \sqrt{41}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 \sqrt{41}$ .

$\cos (θ) = \frac{2 \cdot - 4}{2 (\sqrt{41})}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot - 4}{2 \sqrt{41}}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{- 4}{\sqrt{41}}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{4}{\sqrt{41}}$

Step 4.3

Multiply $\frac{4}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\cos (θ) = - (\frac{4}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}})$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{4}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\cos (θ) = - \frac{4 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 4.4.2

Raise $\sqrt{41}$ to the power of $1$ .

$\cos (θ) = - \frac{4 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 4.4.3

Raise $\sqrt{41}$ to the power of $1$ .

$\cos (θ) = - \frac{4 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 4.4.4

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

$\cos (θ) = - \frac{4 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{4 \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{4 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\cos (θ) = - \frac{4 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\cos (θ) = - \frac{4 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{4 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{4 \sqrt{41}}{41}$

Step 4.4.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{4 \sqrt{41}}{41}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{4 \sqrt{41}}{41} \approx - 0.62469504$