Trigonometry Examples

$(\sqrt{6}, - \sqrt{2})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\sqrt{6}, - \sqrt{2})$ , draw the triangle between the three points $(0, 0)$ , $(\sqrt{6}, 0)$ , and $(\sqrt{6}, - \sqrt{2})$ .

Opposite : $- \sqrt{2}$

Adjacent : $\sqrt{6}$

Step 2

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

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Step 2.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{{(6^{\frac{1}{2}})}^{2} + {(- \sqrt{2})}^{2}}$

Step 2.1.2

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

$\sqrt{6^{\frac{1}{2} \cdot 2} + {(- \sqrt{2})}^{2}}$

Step 2.1.3

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

$\sqrt{6^{\frac{2}{2}} + {(- \sqrt{2})}^{2}}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\sqrt{6^{\frac{2}{2}} + {(- \sqrt{2})}^{2}}$

Step 2.1.4.2

Rewrite the expression.

$\sqrt{6^{1} + {(- \sqrt{2})}^{2}}$

Step 2.1.5

Evaluate the exponent.

$\sqrt{6 + {(- \sqrt{2})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $- \sqrt{2}$ .

$\sqrt{6 + {(- 1)}^{2} {\sqrt{2}}^{2}}$

Step 2.2.2

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

$\sqrt{6 + 1 {\sqrt{2}}^{2}}$

Step 2.2.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$\sqrt{6 + {\sqrt{2}}^{2}}$

Step 2.3

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

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Step 2.3.1

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

$\sqrt{6 + {(2^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\sqrt{6 + 2^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\sqrt{6 + 2^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{6 + 2^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{6 + 2^{1}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{6 + 2}$

Step 2.4

Add $6$ and $2$ .

$\sqrt{8}$

Step 2.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.5.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 2.5.2

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

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

Step 2.6

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 3

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

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

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Combine $\sqrt{6}$ and $\sqrt{2}$ into a single radical.

$\cos (θ) = \frac{\sqrt{\frac{6}{2}}}{2}$

Step 4.2

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

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Step 4.2.1

Factor $2$ out of $6$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.2.2.4

Divide $3$ by $1$ .

$\cos (θ) = \frac{\sqrt{3}}{2}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{3}}{2} \approx 0.8660254$