Answer
See below
Work Step by Step
Trigonometric ratios are the ratio which is used to define the relationship between the lengths of the sides of the right triangle with the angles formed.
There are three fundamental type of trigonometric ratios which are sine ratio, cosine ratio and tan ratio. The longest side of the right triangle is known as hypotenuse. There is an acute angle and a right angle in right angle triangle. The two legs of the triangle are related to an acute angle.
One leg is known as side opposite to the acute angle and other leg is side adjacent to the acute angle. Angle of elevation and the angle of depression are formed by the horizontal line and the line of sight above and below the observer respectively.
The trigonometric ratio can be used to solve the navigation problem when the captain of the ship goes off track.
The length of the side opposite to the acute angle can be measured as using sine ratio:
\[\begin{align}
& \sin \theta =\frac{a}{5\text{ km}} \\
& \sin 15{}^\circ =\frac{a}{5\text{ km}} \\
& a=5\times \sin 15{}^\circ \\
& =5\times 0.258
\end{align}\]
\[=1.29\text{ km}\]
Compute the length of the side adjacent to the acute angle using cosine ratio as follows:
\[\begin{align}
& cos\theta =\frac{b}{5\text{ km}} \\
& cos15{}^\circ =\frac{b}{5\text{ km}} \\
& b=5\text{ km}\times cos15{}^\circ \\
& =5\text{ km}\times 0.965 \\
& =4.825\text{ km}
\end{align}\]
Compute Z, the distance from the current location as follows:
\[\begin{align}
& z=10\text{ km}-b \\
& =10\text{ km}-4.825\text{ km} \\
& =5.175\text{km}
\end{align}\].
