Question

A submarine dives from the water surface at an angle of 30⁰ below the horizontal, following a straight path 50m long. How far is the submarine then below the water surface?

(a) 50m(b) 50m/sin30⁰(c) (50m)sin30⁰(d) (50m)cos30⁰(e) none of those answers.

Short answer

The correct option is (c).

Step-by-step solution

Definition of sine and cosine

The sine and cosine of an angle are trigonometric functions. In the context of a right triangle, the sine and cosine of an acute angle are defined as follows: for the specified angle, the sine is the ratio of the length of the opposite side to the length of the triangle's longest side (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to the hypotenuse.

Explanation for the correct option

As per the above figure and the definition of sine and cosine, $cos\theta =\frac{A_x}{A}andsin\theta =\frac{A_y}{A}.$

Hence the components of $\overrightarrow{A}$are localid="1663659516194" $A_x=Acos\theta and{A}_y=Asin\theta .$

A submarine takes a straight course 50m long from the water's surface at an angle of 30⁰ below the horizontal.

The submarine below the water surface is calculated as

$$\begin{gathered} \sin 30°=\frac{y}{50m} \\ y=(50\hspace{0.33em}m)\sin 30° \end{gathered}$$

So option (c) is correct.

Explanation for option (a)

By calculation it is known that the submarine is 50m(sin30⁰) below the water's surface.

So, option (a) 50m is incorrect.

Explanation for option (b)

By calculation it is known that the submarine is (50m)sin30⁰ below the water's surface.

So, option (b) 50m/sin30⁰ is incorrect.

Explanation for option (d)

By calculation it is known that the submarine is (50m)sin30⁰ below the water's surface.

So, option (d) (50m)sin30⁰ is incorrect.

Explanation for option (e)

By calculation it is known that the submarine is 50m sin30⁰ below the water's surface.

Since, option (c) is correct, which implies that option (e) is incorrect