Find the Cosine Given the Point (2,- square root of 5)

$(2, - \sqrt{5})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(2, - \sqrt{5})$ , draw the triangle between the three points $(0, 0)$ , $(2, 0)$ , and $(2, - \sqrt{5})$ .

Opposite : $- \sqrt{5}$

Adjacent : $2$

Step 2

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

Simplify the expression.

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

$\sqrt{4 + {(- \sqrt{5})}^{2}}$

Apply the product rule to $- \sqrt{5}$ .

$\sqrt{4 + {(- 1)}^{2} {\sqrt{5}}^{2}}$

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

$\sqrt{4 + 1 {\sqrt{5}}^{2}}$

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$\sqrt{4 + {\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{4 + {(5^{\frac{1}{2}})}^{2}}$

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

$\sqrt{4 + 5^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{4 + 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{4 + 5^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{4 + 5^{1}}$

Evaluate the exponent.

$\sqrt{4 + 5}$

Simplify the expression.

Add $4$ and $5$ .

$\sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

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

$3$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{2}{3}$ .

$\frac{2}{3}$

Step 4

Approximate the result.

$\cos (θ) = \frac{2}{3} \approx 0.\overline{6}$

Find the Cosine Given the Point (2,- square root of 5)

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