Find the Cosine Given the Point (28/53,45/53)

$(\frac{28}{53}, \frac{45}{53})$

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{28}{53}, \frac{45}{53})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{28}{53}, 0)$ , and $(\frac{28}{53}, \frac{45}{53})$ .

Opposite : $\frac{45}{53}$

Adjacent : $\frac{28}{53}$

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

Apply the product rule to $\frac{28}{53}$ .

$\sqrt{\frac{28^{2}}{53^{2}} + {(\frac{45}{53})}^{2}}$

Raise $28$ to the power of $2$ .

$\sqrt{\frac{784}{53^{2}} + {(\frac{45}{53})}^{2}}$

Raise $53$ to the power of $2$ .

$\sqrt{\frac{784}{2809} + {(\frac{45}{53})}^{2}}$

Apply the product rule to $\frac{45}{53}$ .

$\sqrt{\frac{784}{2809} + \frac{45^{2}}{53^{2}}}$

Raise $45$ to the power of $2$ .

$\sqrt{\frac{784}{2809} + \frac{2025}{53^{2}}}$

Raise $53$ to the power of $2$ .

$\sqrt{\frac{784}{2809} + \frac{2025}{2809}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{784 + 2025}{2809}}$

Add $784$ and $2025$ .

$\sqrt{\frac{2809}{2809}}$

Divide $2809$ by $2809$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{28}{53}}{1}$ .

$\frac{\frac{28}{53}}{1}$

Divide $\frac{28}{53}$ by $1$ .

$\cos (θ) = \frac{28}{53}$

Approximate the result.

$\cos (θ) = \frac{28}{53} \approx 0.52830188$

Find the Cosine Given the Point (28/53,45/53)

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