Trigonometry Examples

$(- 6, 4)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 6, 4)$ , draw the triangle between the three points $(0, 0)$ , $(- 6, 0)$ , and $(- 6, 4)$ .

Opposite : $4$

Adjacent : $- 6$

Step 2

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

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Step 2.1

Raise $- 6$ to the power of $2$ .

$\sqrt{36 + {(4)}^{2}}$

Step 2.2

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

$\sqrt{36 + 16}$

Step 2.3

Add $36$ and $16$ .

$\sqrt{52}$

Step 2.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.4.1

Factor $4$ out of $52$ .

$\sqrt{4 (13)}$

Step 2.4.2

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

$\sqrt{2^{2} \cdot 13}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{13}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 6}{2 \sqrt{13}}$ .

$\frac{- 6}{2 \sqrt{13}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $- 6$ and $2$ .

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Step 4.1.1

Factor $2$ out of $- 6$ .

$\cos (θ) = \frac{2 \cdot - 3}{2 \sqrt{13}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 \sqrt{13}$ .

$\cos (θ) = \frac{2 \cdot - 3}{2 (\sqrt{13})}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot - 3}{2 \sqrt{13}}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{- 3}{\sqrt{13}}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{3}{\sqrt{13}}$

Step 4.3

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = - (\frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = - \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = - \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = - \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.4

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

$\cos (θ) = - \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\cos (θ) = - \frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\cos (θ) = - \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

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.

$\cos (θ) = - \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{3 \sqrt{13}}{13}$

Step 4.4.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{3 \sqrt{13}}{13}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{3 \sqrt{13}}{13} \approx - 0.83205029$