Trigonometry Examples

$(- \sqrt{11}, - 5)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \sqrt{11}, - 5)$ , draw the triangle between the three points $(0, 0)$ , $(- \sqrt{11}, 0)$ , and $(- \sqrt{11}, - 5)$ .

Opposite : $- 5$

Adjacent : $- \sqrt{11}$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $- \sqrt{11}$ .

$\sqrt{{(- 1)}^{2} {\sqrt{11}}^{2} + {(- 5)}^{2}}$

Step 2.1.2

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

$\sqrt{1 {\sqrt{11}}^{2} + {(- 5)}^{2}}$

Step 2.1.3

Multiply ${\sqrt{11}}^{2}$ by $1$ .

$\sqrt{{\sqrt{11}}^{2} + {(- 5)}^{2}}$

Step 2.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{{(11^{\frac{1}{2}})}^{2} + {(- 5)}^{2}}$

Step 2.2.2

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

$\sqrt{11^{\frac{1}{2} \cdot 2} + {(- 5)}^{2}}$

Step 2.2.3

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

$\sqrt{11^{\frac{2}{2}} + {(- 5)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{11^{\frac{2}{2}} + {(- 5)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{11^{1} + {(- 5)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{11 + {(- 5)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Raise $- 5$ to the power of $2$ .

$\sqrt{11 + 25}$

Step 2.3.2

Add $11$ and $25$ .

$\sqrt{36}$

Step 2.3.3

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

Step 2.3.4

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

$6$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \sqrt{11}}{6}$ .

$\frac{- \sqrt{11}}{6}$

Step 4

Move the negative in front of the fraction.

$\cos (θ) = - \frac{\sqrt{11}}{6}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{\sqrt{11}}{6} \approx - 0.55277079$