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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.10

Question 1.

Determine whether the following measurements produce one triangle, two triangles or no triangle:

∠B = 88°, a = 23, b = 2. Solve if solution exists.

Solution:

We are given a = 23,

b = 2, and

∠B = 88°.

So we can

Question 2.

If the sides of a ∆ABC are a = 4, b = 6 and c = 8, then show that 4 cos B + 3 cos C = 2.

Solution:

a = 4,

b = 6,

c = 8

To prove 4 cos B + 3 cos C = 2

Question 3.

In a ∆ABC, if a = \(\sqrt{3}\) – 1, b = \(\sqrt{3}\) + 1 and C = 60°, find the other side and other two angles.

Solution:

Question 4.

Solution:

Question 5.

In a ∆ABC, if a = 12 cm, b = 8 cm and C = 30°, then show that its area is 24 sq.cm.

Solution:

a = 12 cm,

b = 8 cm,

C = 30°

Question 6.

In a ∆ABC, if a = 18 cm, b = 24 cm and c = 30 cm, then show that its area is 216 sq.cm.

Solution:

a = 18 cm,

b = 24 cm,

c = 30 cm

The sides form a right angled triangle

Question 7.

Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30° and 45° respectively. If A and B stand 5 km apart, find the distance of the intruder from B.

Solution:

By using sine formula

\(\frac{x}{\sin 30^{\circ}}=\frac{5}{\sin 15^{\circ}}\)

Question 8.

A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point P, he finds the distance to the eastern-most point of the pond to be 8 km, while the distance to the western most point from P to be 6 km. If the angle between the two lines of sight is 60°, find the width of the pond.

Solution:

p^{2} = W^{2} + E^{2} – 2WE cos P

P^{2} = 64 + 36 – 2 × 8 × 6 × Cos 60°

Question 9.

Two Navy helicopters A and B are flying over the Bay of Bengal at same altitude from the sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60° at the boat, find the distance of the boat from B.

Solution:

Question 10.

A straight tunnel is to be made through a mountain. A surveyor observes the two extremities A and B of the tunnel to be built from a point P in front of the mountain. If AP = 3 km, BP = 5 km and ∠APB = 120°, then find the length of the tunnel to be built.

Solution:

A, B are the two extremities of the tunnel.

P – Point of observation.

PA , PB are the directions of the points A, B as observed from the point P

AP = 3 km, BP = 5 km, ∠ APB = 120°

Using cosine formula in ∆ APB

AB^{2} = AP^{2} + BP^{2} – 2AP. BP. cos(120°)

AB^{2} = 3^{2} + 5^{2} – 2 × 3 × 5 cos (180° – 60°)

= 9 + 25 – 30 (- cos 60°)

= 34 + 30 × \(\frac { 1 }{ 2 }\)

= 34 + 15 = 49

AB = √49 = 7

∴ The length of the tunnel AB = 7 k.m.

Question 11.

A farmer wants to purchase a triangular-shaped land with sides 120 feet and 60 feet and the angle included between these two sides is 60°. If the land costs ₹ 500 per sq. ft, find the amount he needed to purchase the land. Also, find the perimeter of the land.

Solution:

Question 12.

A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30°. If after 100 km, the target has an angle of depression of 60°, how far is the target from the fighter jet at that instant?

Solution:

Question 13.

A plane is 1 km from one landmark and 2 km from another. From the plane’s point of view, the land between them subtends an angle of 60°. How far apart are the landmarks?

Solution:

Question 14.

A man starts his morning walk at a point A reaches two points B and C and finally back to A such that ∠A= 60° and ∠B = 45°, AC = 4 km in the ∆ABC. Find the total distance he covered during his morning walk.

Solution:

Question 15.

Two vehicles leave the same place P at the same time moving along two different roads. One vehicle moves at an average speed of 60 km/hr and the other vehicle moves at an average speed of 80 km/hr. After half an hour the vehicle reaches destinations A and B. If AB subtends 60° at the initial point P, then find AB.

Solution:

= 900+ 1600 – 1200 = 1300

Question 16.

Suppose that a satellite in space, an earth station, and the centre of earth all lie in the same plane. Let r be the radius of earth and R be the distance from the centre of the earth to the satellite. Let d be the distance from the earth station to the satellite. Let 30° be the angle of elevation from the earth station to the satellite. If the line segment connecting earth station and satellite subtends angle α at the centre of the earth, then prove that

Solution:

### Samacheer Kalvi 11th Maths Solutions Chapter 3 Trigonometry Ex 3.10 Additional Questions Solved

Question 1.

Given a = 8, b = 9, c = 10, find all the angles.

Solution:

Question 2.

Given a = 31, b = 42, c = 57, find all the angles.

Solution:

Since the sides are larger quantities, use half angles formulae

Question 3.

In a triangle ABC, A = 35° 17′ ; C = 45° 13′ ; b = 42.1 Solve the triangle

Solution:

The unknown parts are B, a, c,

B = 180 – (A + C) = 180 – (35° 17′ + 45° 13′)

= 99° 30′

To find sides, use sine formula

log c = log 42.1 + log sin 45° 31 – log sin 99° 30′

= 1.6243 + 1.8511 – 1.9940

= 1.4754 – 1.9940

= 1.4754 – [-1 + 0.9940] = 1.4814

⇒ c = 30.3°

Thus B = 99° 30′ ; a = 24.65° ; c = 30.3°

Question 4.

Solve the triangle ABC if a = 5, b = 4 and C = 68°.

Solution:

To find c, use c^{2} = a^{2} + b^{2} – 2ab cos C

c^{2} = 25 + 16 – 2 × 5 × 4 cos 68°

= 41 – 40 × 0.3746 = 26.016

c = 5.1

To find the other two angles, use sine formula.