Question

how far would you now have to sail?
a. 0.68 miles
b. 0.43 miles
c. 0.25 miles
d. 0.5 miles

Explanation:

Step1: Identify the triangle type

We have a right - triangle with hypotenuse \(h = 0.5\) miles and an angle \(\theta=60^{\circ}\). We want to find the side opposite the \(60^{\circ}\) angle.

Step2: Use the sine function

The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\), and \(h = 0.5\) miles. Let the side we want to find be \(x\). So, \(\sin60^{\circ}=\frac{x}{0.5}\).

Step3: Solve for \(x\)

Cross - multiply to get \(x = 0.5\times\sin60^{\circ}\). Substituting \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(x=0.5\times\frac{\sqrt{3}}{2}\approx0.5\times0.866 = 0.433\approx0.43\) miles.

Answer:

b. 0.43 miles