Question

Skysails’ is that genre of engineering science that uses extensive utilization of wind energy to move a vessel in the sea water. The ‘Skysails’ technology allows the towing kite to gain a height of anything between 100 metres – 300 metres. The sailing kite is made in such a way that it can be raised to its proper elevation and then brought back with the help of a ‘telescopic mast’ that enables the kite to be raised properly and effectively. Based on the following figure related to sky sailing, answer the questions: (i) In the given figure, if sin = cos (3−30), where and 3−30 are acute angles, then find the value of . (ii) What should be the length of the rope of the kite sail in order to pull the ship at the angle (calculated above) and be at a vertical height of 200 m? 2

Answer 1

Step-by-step explanation:

We know that from complementary angle formula

$\sin( \theta) = \cos(90 - \theta)$

(i) In the given figure, if sin theta = cos (3theta−30), where and 3theta−30 are acute angles, then find the value of theta.

From complementary angle formula

$\sin( \theta) = \cos( 3\theta - 30°) \\ \\ \cos(90° - \theta) = \cos( 3\theta - 30°) \\ \\90° - \theta = 3\theta - 30° \\ \\ - 4\theta = - 30° - 90° \\ \\ 4\theta = 120° \\ \\ \theta = \frac{120°}{4} \\ \\ \theta = 30°$

(ii) What should be the length of the rope of the kite sail in order to pull the ship at the angle (calculated above) and be at a vertical height of 200 m.

Since angle is 30°,to find the length of rope apply trigonometric identity

$\sin( \theta) = \frac{AB}{AC} \\ \\ \sin( 30°) = \frac{200}{AC} \\ \\ \frac{1}{2} = \frac{200}{AC} \\ \\ AC= 400 \: m$

Hope it helps you.