Find the Cosine Given the Point (( square root of 7)/4,3/4)

$(\frac{\sqrt{7}}{4}, \frac{3}{4})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{7}}{4}, \frac{3}{4})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{7}}{4}, 0)$ , and $(\frac{\sqrt{7}}{4}, \frac{3}{4})$ .

Opposite : $\frac{3}{4}$

Adjacent : $\frac{\sqrt{7}}{4}$

Step 2

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

Apply the product rule to $\frac{\sqrt{7}}{4}$ .

$\sqrt{\frac{{\sqrt{7}}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{\frac{{(7^{\frac{1}{2}})}^{2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

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

$\sqrt{\frac{7^{\frac{1}{2} \cdot 2}}{4^{2}} + {(\frac{3}{4})}^{2}}$

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

$\sqrt{\frac{7^{\frac{2}{2}}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{7^{\frac{2}{2}}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{7^{1}}{4^{2}} + {(\frac{3}{4})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{7}{4^{2}} + {(\frac{3}{4})}^{2}}$

Simplify the expression.

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

$\sqrt{\frac{7}{16} + {(\frac{3}{4})}^{2}}$

Apply the product rule to $\frac{3}{4}$ .

$\sqrt{\frac{7}{16} + \frac{3^{2}}{4^{2}}}$

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

$\sqrt{\frac{7}{16} + \frac{9}{4^{2}}}$

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

$\sqrt{\frac{7}{16} + \frac{9}{16}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{7 + 9}{16}}$

Add $7$ and $9$ .

$\sqrt{\frac{16}{16}}$

Divide $16$ by $16$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{\sqrt{7}}{4}}{1}$ .

$\frac{\frac{\sqrt{7}}{4}}{1}$

Step 4

Divide $\frac{\sqrt{7}}{4}$ by $1$ .

$\cos (θ) = \frac{\sqrt{7}}{4}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{7}}{4} \approx 0.66143782$

Find the Cosine Given the Point (( square root of 7)/4,3/4)

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