Precalculus Examples

$(\frac{3}{7}, - \frac{2 \sqrt{10}}{7})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3}{7}, - \frac{2 \sqrt{10}}{7})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3}{7}, 0)$ , and $(\frac{3}{7}, - \frac{2 \sqrt{10}}{7})$ .

Opposite : $- \frac{2 \sqrt{10}}{7}$

Adjacent : $\frac{3}{7}$

Step 2

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

Tap for more steps...

Step 2.1

Apply the product rule to $\frac{3}{7}$ .

$\sqrt{\frac{3^{2}}{7^{2}} + {(- \frac{2 \sqrt{10}}{7})}^{2}}$

Step 2.2

Raise $3$ to the power of $2$ .

$\sqrt{\frac{9}{7^{2}} + {(- \frac{2 \sqrt{10}}{7})}^{2}}$

Step 2.3

Raise $7$ to the power of $2$ .

$\sqrt{\frac{9}{49} + {(- \frac{2 \sqrt{10}}{7})}^{2}}$

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.4.1

Apply the product rule to $- \frac{2 \sqrt{10}}{7}$ .

$\sqrt{\frac{9}{49} + {(- 1)}^{2} {(\frac{2 \sqrt{10}}{7})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{2 \sqrt{10}}{7}$ .

$\sqrt{\frac{9}{49} + {(- 1)}^{2} \frac{{(2 \sqrt{10})}^{2}}{7^{2}}}$

Step 2.4.3

Apply the product rule to $2 \sqrt{10}$ .

$\sqrt{\frac{9}{49} + {(- 1)}^{2} \frac{2^{2} {\sqrt{10}}^{2}}{7^{2}}}$

Step 2.5

Simplify the expression.

Tap for more steps...

Step 2.5.1

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

$\sqrt{\frac{9}{49} + 1 \frac{2^{2} {\sqrt{10}}^{2}}{7^{2}}}$

Step 2.5.2

Multiply $\frac{2^{2} {\sqrt{10}}^{2}}{7^{2}}$ by $1$ .

$\sqrt{\frac{9}{49} + \frac{2^{2} {\sqrt{10}}^{2}}{7^{2}}}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

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

$\sqrt{\frac{9}{49} + \frac{4 {\sqrt{10}}^{2}}{7^{2}}}$

Step 2.6.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 2.6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{\frac{9}{49} + \frac{4 {(10^{\frac{1}{2}})}^{2}}{7^{2}}}$

Step 2.6.2.2

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

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10^{\frac{1}{2} \cdot 2}}{7^{2}}}$

Step 2.6.2.3

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

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10^{\frac{2}{2}}}{7^{2}}}$

Step 2.6.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.2.4.1

Cancel the common factor.

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10^{\frac{2}{2}}}{7^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10^{1}}{7^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10}{7^{2}}}$

Step 2.7

Simplify the expression.

Tap for more steps...

Step 2.7.1

Raise $7$ to the power of $2$ .

$\sqrt{\frac{9}{49} + \frac{4 \cdot 10}{49}}$

Step 2.7.2

Multiply $4$ by $10$ .

$\sqrt{\frac{9}{49} + \frac{40}{49}}$

Step 2.7.3

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 40}{49}}$

Step 2.7.4

Add $9$ and $40$ .

$\sqrt{\frac{49}{49}}$

Step 2.7.5

Divide $49$ by $49$ .

$\sqrt{1}$

Step 2.7.6

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{3}{7}}{1}$ .

$\frac{\frac{3}{7}}{1}$

Step 4

Divide $\frac{3}{7}$ by $1$ .

$\cos (θ) = \frac{3}{7}$

Step 5

Approximate the result.

$\cos (θ) = \frac{3}{7} \approx 0.\overline{428571}$