# RD Sharma solutions for Class 11 Mathematics Textbook chapter 10 - Sine and cosine formulae and their applications [Latest edition]

#### Chapters ## Chapter 10: Sine and cosine formulae and their applications

Exercise 10.1Exercise 10.2Others
Exercise 10.1 [Pages 12 - 14]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 10 Sine and cosine formulae and their applications Exercise 10.1 [Pages 12 - 14]

Exercise 10.1 | Q 1 | Page 12

If in ∆ABC, ∠A = 45°, ∠B = 60° and ∠C = 75°, find the ratio of its sides.

Exercise 10.1 | Q 2 | Page 12

If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b

Exercise 10.1 | Q 3 | Page 12

In ∆ABC, if a = 18, b = 24 and c = 30 and ∠c = 90°, find sin A, sin B and sin C

Exercise 10.1 | Q 4 | Page 12

In triangle ABC, prove the following:

$\frac{a - b}{a + b} = \frac{\tan \left( \frac{A - B}{2} \right)}{\tan \left( \frac{A + B}{2} \right)}$

Exercise 10.1 | Q 5 | Page 13

In triangle ABC, prove the following:

$\left( a - b \right) \cos \frac{C}{2} = c \sin \left( \frac{A - B}{2} \right)$

Exercise 10.1 | Q 6 | Page 13

In triangle ABC, prove the following:

$\frac{c}{a - b} = \frac{\tan\left( \frac{A}{2} \right) + \tan \left( \frac{B}{2} \right)}{\tan \left( \frac{A}{2} \right) - \tan \left( \frac{B}{2} \right)}$

Exercise 10.1 | Q 7 | Page 13

In triangle ABC, prove the following:

$\frac{c}{a + b} = \frac{1 - \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right)}{1 + \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right)}$

Exercise 10.1 | Q 8 | Page 13

In triangle ABC, prove the following:

$\frac{a + b}{c} = \frac{\cos \left( \frac{A - B}{2} \right)}{\sin \frac{C}{2}}$

Exercise 10.1 | Q 9 | Page 13

In any triangle ABC, prove the following:

$\sin \left( \frac{B - C}{2} \right) = \frac{b - c}{a} \cos\frac{A}{2}$

Exercise 10.1 | Q 10 | Page 13

In triangle ABC, prove the following:

$\frac{a^2 - c^2}{b^2} = \frac{\sin \left( A - C \right)}{\sin \left( A + C \right)}$

Exercise 10.1 | Q 11 | Page 13

In triangle ABC, prove the following:

$b \sin B - c \sin C = a \sin \left( B - C \right)$

Exercise 10.1 | Q 12 | Page 13

In triangle ABC, prove the following:

$a^2 \sin \left( B - C \right) = \left( b^2 - c^2 \right) \sin A$

Exercise 10.1 | Q 13 | Page 13

In triangle ABC, prove the following:

$\frac{\sqrt{\sin A} - \sqrt{\sin B}}{\sqrt{\sin A} + \sqrt{\sin B}} = \frac{a + b - 2\sqrt{ab}}{a - b}$

Exercise 10.1 | Q 14 | Page 13

In triangle ABC, prove the following:

$a \left( \sin B - \sin C \right) + \left( \sin C - \sin A \right) + c \left( \sin A - \sin B \right) = 0$

Exercise 10.1 | Q 15 | Page 13

In triangle ABC, prove the following:

$\frac{a^2 \sin \left( B - C \right)}{\sin A} + \frac{b^2 \sin \left( C - A \right)}{\sin B} + \frac{c^2 \sin \left( A - B \right)}{\sin C} = 0$

Exercise 10.1 | Q 16 | Page 13

In triangle ABC, prove the following:

$a^2 \left( \cos^2 B - \cos^2 C \right) + b^2 \left( \cos^2 C - \cos^2 A \right) + c^2 \left( \cos^2 A - \cos^2 B \right) = 0$

Exercise 10.1 | Q 17 | Page 13

In triangle ABC, prove the following:

$b \cos B + c \cos C = a \cos \left( B - C \right)$

Exercise 10.1 | Q 18 | Page 13

In triangle ABC, prove the following:

$\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2} - \frac{1}{a^2} - \frac{1}{b^2}$

Exercise 10.1 | Q 19 | Page 13

In triangle ABC, prove the following:

$\frac{\cos^2 B - \cos^2 C}{b + c} + \frac{\cos^2 C - \cos^2 A}{c + a} + \frac{co s^2 A - \cos^2 B}{a + b} = 0$

Exercise 10.1 | Q 20 | Page 13

In ∆ABC, prove that: $a \sin\frac{A}{2} \sin \left( \frac{B - C}{2} \right) + b \sin \frac{B}{2} \sin \left( \frac{C - A}{2} \right) + c \sin \frac{C}{2} \sin \left( \frac{A - B}{2} \right) = 0$

Exercise 10.1 | Q 21 | Page 13

In ∆ABC, prove that: $\frac{b \sec B + c \sec C}{\tan B + \tan C} = \frac{c \sec C + a \sec A}{\tan C + \tan A} = \frac{a \sec A + b \sec B}{\tan A + \tan B}$

Exercise 10.1 | Q 22 | Page 13

In triangle ABC, prove the following:

$a \cos A + b\cos B + c \cos C = 2b \sin A \sin C = 2 c \sin A \sin B$

Exercise 10.1 | Q 23 | Page 13

$a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)$

Exercise 10.1 | Q 24 | Page 13

In ∆ABC, prove that $a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .$

Exercise 10.1 | Q 25 | Page 13

In ∆ABC, prove that if θ be any angle, then b cosθ = c cos (A − θ) + a cos (C + θ).

Exercise 10.1 | Q 26 | Page 13

In ∆ABC, if sin2 A + sin2 B = sin2 C. show that the triangle is right-angled.

Exercise 10.1 | Q 27 | Page 14

In ∆ABC, if a2b2 and c2 are in A.P., prove that cot A, cot B and cot C are also in A.P.

Exercise 10.1 | Q 28 | Page 14

The upper part of a tree broken by the wind makes an angle of 30° with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.

Exercise 10.1 | Q 29 | Page 14

At the foot of a mountain, the elevation of it summit is 45°; after ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain.

Exercise 10.1 | Q 30 | Page 14

A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c metres along a slope inclined at an angle β and finds the angle of elevation of the peak of the hill to be ϒ. Show that the height of the peak above the ground is $\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}$

Exercise 10.1 | Q 31 | Page 14

If the sides ab and c of ∆ABC are in H.P., prove that $\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}$

Exercise 10.2 [Pages 25 - 26]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 10 Sine and cosine formulae and their applications Exercise 10.2 [Pages 25 - 26]

Exercise 10.2 | Q 1 | Page 25

In $∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°$  show that its area is $\frac{15\sqrt{3}}{2} sq$.units.

Exercise 10.2 | Q 2 | Page 25

In $∆ ABC, if a = \sqrt{2}, b = \sqrt{3} \text{ and } c = \sqrt{5}$ show that its area is $\frac{1}{2}\sqrt{6} sq .$ units.

Exercise 10.2 | Q 3 | Page 25

The sides of a triangle are a = 4, b = 6 and c = 8. Show that $8 \cos A + 16 \cos B + 4 \cos C = 17$

Exercise 10.2 | Q 4 | Page 25

In ∆ ABC, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C

Exercise 10.2 | Q 5 | Page 25

In ∆ABC, prove the following: $b \left( c \cos A - a \cos C \right) = c^2 - a^2$

Exercise 10.2 | Q 6 | Page 25

In ∆ABC, prove the following: $c \left( a \cos B - b \cos A \right) = a^2 - b^2$

Exercise 10.2 | Q 7 | Page 25

In ∆ABC, prove  the following:

$2 \left( bc \cos A + ca \cos B + ab \cos C \right) = a^2 + b^2 + c^2$

Exercise 10.2 | Q 8 | Page 25

In ∆ABC, prove the following

$\left( c^2 - a^2 + b^2 \right) \tan A = \left( a^2 - b^2 + c^2 \right) \tan B = \left( b^2 - c^2 + a^2 \right) \tan C$

Exercise 10.2 | Q 9 | Page 25

In ∆ABC, prove the following:

$\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}$

Exercise 10.2 | Q 10 | Page 25

In ∆ABC, prove that  $a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0$

Exercise 10.2 | Q 11 | Page 25

a cos + b cos B + c cos C = 2sin sin

Exercise 10.2 | Q 12 | Page 25

In ∆ABC, prove the following:

$a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}$

Exercise 10.2 | Q 13 | Page 25

In ∆ABC, prove the following:

$4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2$

Exercise 10.2 | Q 14 | Page 25

In ∆ABC, prove the following:

$\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C$

Exercise 10.2 | Q 15 | Page 25

In $∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}$  Prove that $\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}$

Exercise 10.2 | Q 16 | Page 25

In $∆ ABC, if \angle B = 60°,$  prove that $\left( a + b + c \right) \left( a - b + c \right) = 3ca$

Exercise 10.2 | Q 17 | Page 25

If in $∆ ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1$ prove that the triangle is right-angled.

Exercise 10.2 | Q 18 | Page 25

In $∆ ABC \text{ if } \cos C = \frac{\sin A}{2 \sin B}$ prove that the triangle is isosceles.

Exercise 10.2 | Q 19 | Page 26

Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38° E and other travels 32 km/hr in the direction S 52° E. Find the distance between the ships at the end of 3 hrs.

[Page 26]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 10 Sine and cosine formulae and their applications [Page 26]

Q 1 | Page 26

Answer  the following questions in one word or one sentence or as per exact requirement of the question.

Find the area of the triangle ∆ABC in which a = 1, b = 2 and $\angle C = 60º$

Q 2 | Page 26

Answer  the following questions in one word or one sentence or as per exact requirement of the question.In a ∆ABC, if b =$\sqrt{3}$ and $\angle A = 30°$  find a

Q 3 | Page 26

Answer  the following questions in one word or one sentence or as per exact requirement of the question.

In a ∆ABC, if $\cos A = \frac{\sin B}{2\sin C}$  then show that c = a

Q 4 | Page 26

Answer  the following questions in one word or one sentence or as per exact requirement of the question.

In a ∆ABC, if b = 20, c = 21 and $\sin A = \frac{3}{5}$

Q 5 | Page 26

Answer  the following questions in one word or one sentence or as per exact requirement of the question.

In a ∆ABC, if sinA and sinB are the roots of the equation  $c^2 x^2 - c\left( a + b \right)x + ab = 0$  then find $\angle C$

Q 6 | Page 26

Answer the following questions in one word or one sentence or as per exact requirement of the question.

In ∆ABC, if a = 8, b = 10, c = 12 and C = λA, find the value of λ

Q 7 | Page 26

Answer the following questions in one word or one sentence or as per exact requirement of the question.

If the sides of a triangle are proportional to 2, $\sqrt{6}$ and $\sqrt{3} - 1$ find the measure of its greatest angle.

Q 8 | Page 26

Answer the following questions in one word or one sentence or as per exact requirement of the question.

If in a ∆ABC, $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$ then find the measures of angles ABC

Q 9 | Page 26

Answer the following questions in one word or one sentence or as per exact requirement of the question.

In any triangle ABC, find the value of $a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\ Q 10 | Page 26 Answer the following questions in one word or one sentence or as per exact requirement of the question. In any ∆ABC, find the value of \[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)$

[Pages 26 - 27]

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 10 Sine and cosine formulae and their applications [Pages 26 - 27]

Q 1 | Page 26

Mark the correct alternative in each of the following:
In any ∆ABC, $\sum^{}_{} a^2 \left( \sin B - \sin C \right)$ =

• $a^2 + b^2 + c^2$

• $a^2$

• $b^2$

•  0

Q 2 | Page 26

Mark the correct alternative in each of the following:

In a ∆ABC, if a = 2, $\angle B = 60°$  and$\angle C = 75°$

• $\sqrt{3}$

• $\sqrt{6}$

• $\sqrt{9}$

• $1 + \sqrt{2}$

Q 3 | Page 26

Mark the correct alternative in each of the following:
If the sides of a triangle are in the ratio $1: \sqrt{3}: 2$ then the measure of its greatest angle is

• $\frac{\pi}{6}$

• $\frac{\pi}{3}$

• $\frac{\pi}{2}$

• $\frac{2\pi}{3}$

Q 4 | Page 26

Mark the correct alternative in each of the following:

In any ∆ABC, 2(bc cosA + ca cosB + ab cosC) =

• $abc$

• $a + b + c$

• $a^2 + b^2 + c^2$

• $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$

Q 5 | Page 27

Mark the correct alternative in each of the following:

In a triangle ABC, a = 4, b = 3, $\angle A = 60°$   then c is a root of the equation

• $c^2 - 3c - 7 = 0$

• $c^2 + 3c + 7 = 0$

• $c^2 - 3c + 7 = 0$

• $c^2 + 3c - 7 = 0$

Q 6 | Page 27

Mark the correct alternative in each of the following:

In a ∆ABC, if  $\left( c + a + b \right)\left( a + b - c \right) = ab$ then the measure of angle C is

• $\frac{\pi}{3}$

• $\frac{\pi}{6}$

• $\frac{2\pi}{3}$

• $\frac{\pi}{2}$

Q 7 | Page 27

Mark the correct alternative in each of the following:

In any ∆ABC, the value of  $2ac\sin\left( \frac{A - B + C}{2} \right)$  is

• $a^2 + b^2 - c^2$

• $c^2 + a^2 - b^2$

• $b^2 - c^2 - a^2$

• $c^2 - a^2 - b^2$

Q 8 | Page 27

Mark the correct alternative in each of the following:

In any ∆ABC, $a\left( b\cos C - c\cos B \right) =$

• $a^2$

• $b^2 - c^2$

• 0

• $b^2 + c^2$

## Chapter 10: Sine and cosine formulae and their applications

Exercise 10.1Exercise 10.2Others ## RD Sharma solutions for Class 11 Mathematics Textbook chapter 10 - Sine and cosine formulae and their applications

RD Sharma solutions for Class 11 Mathematics Textbook chapter 10 (Sine and cosine formulae and their applications) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 10 Sine and cosine formulae and their applications are Transformation Formulae, Values of Trigonometric Functions at Multiples and Submultiples of an Angle, Sine and Cosine Formulae and Their Applications, 180 Degree Plusminus X Function, 2X Function, 3X Function, Expressing Sin (X±Y) and Cos (X±Y) in Terms of Sinx, Siny, Cosx and Cosy and Their Simple Applications, Concept of Angle, Introduction of Trigonometric Functions, Signs of Trigonometric Functions, Domain and Range of Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles, Trigonometric Equations, Truth of the Identity, Negative Function Or Trigonometric Functions of Negative Angles, 90 Degree Plusminus X Function, Conversion from One Measure to Another, Graphs of Trigonometric Functions.

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