Find the Sine Given the Point (3/8,( square root of 55)/8)

$(\frac{3}{8}, \frac{\sqrt{55}}{8})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3}{8}, \frac{\sqrt{55}}{8})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3}{8}, 0)$ , and $(\frac{3}{8}, \frac{\sqrt{55}}{8})$ .

Opposite : $\frac{\sqrt{55}}{8}$

Adjacent : $\frac{3}{8}$

Step 2

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

Simplify the expression.

Apply the product rule to $\frac{3}{8}$ .

$\sqrt{\frac{3^{2}}{8^{2}} + {(\frac{\sqrt{55}}{8})}^{2}}$

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

$\sqrt{\frac{9}{8^{2}} + {(\frac{\sqrt{55}}{8})}^{2}}$

Raise $8$ to the power of $2$ .

$\sqrt{\frac{9}{64} + {(\frac{\sqrt{55}}{8})}^{2}}$

Apply the product rule to $\frac{\sqrt{55}}{8}$ .

$\sqrt{\frac{9}{64} + \frac{{\sqrt{55}}^{2}}{8^{2}}}$

Rewrite ${\sqrt{55}}^{2}$ as $55$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{55}$ as $55^{\frac{1}{2}}$ .

$\sqrt{\frac{9}{64} + \frac{{(55^{\frac{1}{2}})}^{2}}{8^{2}}}$

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

$\sqrt{\frac{9}{64} + \frac{55^{\frac{1}{2} \cdot 2}}{8^{2}}}$

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

$\sqrt{\frac{9}{64} + \frac{55^{\frac{2}{2}}}{8^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{9}{64} + \frac{55^{\frac{2}{2}}}{8^{2}}}$

Rewrite the expression.

$\sqrt{\frac{9}{64} + \frac{55^{1}}{8^{2}}}$

Evaluate the exponent.

$\sqrt{\frac{9}{64} + \frac{55}{8^{2}}}$

Simplify the expression.

Raise $8$ to the power of $2$ .

$\sqrt{\frac{9}{64} + \frac{55}{64}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 55}{64}}$

Add $9$ and $55$ .

$\sqrt{\frac{64}{64}}$

Divide $64$ by $64$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{\sqrt{55}}{8}}{1}$ .

$\frac{\frac{\sqrt{55}}{8}}{1}$

Step 4

Divide $\frac{\sqrt{55}}{8}$ by $1$ .

$\sin (θ) = \frac{\sqrt{55}}{8}$

Step 5

Approximate the result.

$\sin (θ) = \frac{\sqrt{55}}{8} \approx 0.92702481$

Find the Sine Given the Point (3/8,( square root of 55)/8)

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