Some Applications of Trigonometry | Exercise 9.1

Question 1

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is $30^\circ$ (see Fig. 9.11).

Solution

Understand the Question

The vertical pole and the ground form a right-angled triangle $\text{ABC}$ where $\angle \text{B} = 90^\circ$ .
The rope $\text{AC} = 20\text{ m}$ acts as the hypotenuse, and the pole $\text{AB}$ is the side opposite to the angle made with the ground ( $\angle \text{C} = 30^\circ$ ).
To find the height of the pole ( $\text{AB}$ ), we use the sine trigonometric ratio: $\sin \theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$ .

Step 1 · Find the Height of the Pole

Let $\text{AB}$ be the vertical pole and $\text{AC}$ be the rope. Diagram 1

In right-angled triangle $\text{ABC}$ $\sin \text{C} = \dfrac{\text{AB}}{\text{AC}}$

Substitute $\text{C} = 30^\circ$ and $\text{AC} = 20\text{ m}$ $\sin 30^\circ = \dfrac{\text{AB}}{20}$

Since $\sin 30^\circ = \dfrac{1}{2}$ $\dfrac{1}{2} = \dfrac{\text{AB}}{20}$

$\text{AB} = 20 \times \dfrac{1}{2}$

$\text{AB} = 10\text{ m}$

Answer

$10\text{ m}$

Common Mistakes

Wrong Trigonometric Ratio: Using $\cos 30^\circ = \dfrac{\text{BC}}{\text{AC}}$ or $\tan 30^\circ = \dfrac{\text{AB}}{\text{BC}}$ instead of $\sin 30^\circ$ , which directly relates the unknown opposite side to the known hypotenuse.
Incorrect Value of $\sin 30^\circ$ : Confusing $\sin 30^\circ = \dfrac{1}{2}$ with $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$ or $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$ .

More questions in Exercise 9.1

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of $30^\circ$ to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m, and inclined at an angle of $60^\circ$ to the ground. What should be the length of the slide in each case?

The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$ . Find the height of the tower.

A kite is flying at a height of $60\text{ m}$ above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60^\circ$ . Find the length of the string, assuming that there is no slack in the string.

A $1.5\text{ m}$ tall boy is standing at some distance from a $30\text{ m}$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^\circ$ to $60^\circ$ as he walks towards the building. Find the distance he walked towards the building.

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^\circ$ and $60^\circ$ respectively. Find the height of the tower.

A statue, $1.6\text{ m}$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point the angle of elevation of the top of the pedestal is $45^\circ$ . Find the height of the pedestal.

The angle of elevation of the top of a building from the foot of the tower is $30^\circ$ and the angle of elevation of the top of the tower from the foot of the building is $60^\circ$ . If the tower is 50 m high, find the height of the building.

Q10

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$ , respectively. Find the height of the poles and the distances of the point from the poles.

Q11

A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^\circ$ . From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is $30^\circ$ (see Fig. 9.12). Find the height of the tower and the width of the canal.

Q12

From the top of a $7\text{ m}$ high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$ . Determine the height of the tower.

Q13

As observed from the top of a $75\text{ m}$ high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$ . If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Q14

A $1.2\text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\text{ m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$ . After some time, the angle of elevation reduces to $30^\circ$ (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Q15

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$ , which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60^\circ$ . Find the time taken by the car to reach the foot of the tower from this point.