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# RD Sharma solutions for Class 11 Mathematics chapter 10 - Sine and cosine formulae and their applications

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

Ex. 10.10Ex. 10.20Others

#### Chapter 10: Sine and cosine formulae and their applications Exercise 10.10 solutions [Pages 12 - 14]

Ex. 10.10 | Q 1 | Page 12

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

Ex. 10.10 | Q 2 | Page 12

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

Ex. 10.10 | 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

Ex. 10.10 | Q 4 | Page 12

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 5 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 6 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 7 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 8 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 9 | Page 13

In any triangle ABC, prove the following:

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

Ex. 10.10 | Q 10 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 11 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 12 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 13 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 14 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 15 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 16 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 17 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 18 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 19 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 20 | Page 13

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

Ex. 10.10 | Q 21 | Page 13

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

Ex. 10.10 | Q 22 | Page 13

In triangle ABC, prove the following:

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

Ex. 10.10 | Q 23 | Page 13

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

Ex. 10.10 | Q 24 | Page 13

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

Ex. 10.10 | Q 25 | Page 13

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

Ex. 10.10 | Q 26 | Page 13

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

Ex. 10.10 | 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.

Ex. 10.10 | 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.

Ex. 10.10 | 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.

Ex. 10.10 | 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 $\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}$

Ex. 10.10 | Q 31 | Page 14

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

#### Chapter 10: Sine and cosine formulae and their applications Exercise 10.20 solutions [Pages 25 - 26]

Ex. 10.20 | 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.

Ex. 10.20 | 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.

Ex. 10.20 | Q 3 | Page 25

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

Ex. 10.20 | Q 4 | Page 25

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

Ex. 10.20 | Q 5 | Page 25

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

Ex. 10.20 | Q 6 | Page 25

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

Ex. 10.20 | Q 7 | Page 25

In ∆ABC, prove  the following:

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

Ex. 10.20 | Q 8 | Page 25

In ∆ABC, prove the following

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

Ex. 10.20 | Q 9 | Page 25

In ∆ABC, prove the following:

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

Ex. 10.20 | Q 10 | Page 25

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

Ex. 10.20 | Q 11 | Page 25

a cos + b cos B + c cos C = 2sin sin

Ex. 10.20 | Q 12 | Page 25

In ∆ABC, prove the following:

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

Ex. 10.20 | Q 13 | Page 25

In ∆ABC, prove the following:

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

Ex. 10.20 | Q 14 | Page 25

In ∆ABC, prove the following:

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

Ex. 10.20 | Q 15 | Page 25

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

Ex. 10.20 | Q 16 | Page 25

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

Ex. 10.20 | Q 17 | Page 25

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

Ex. 10.20 | Q 18 | Page 25

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

Ex. 10.20 | 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.

#### Chapter 10: Sine and cosine formulae and their applications solutions [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)$

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

Ex. 10.10Ex. 10.20Others

## RD Sharma solutions for Class 11 Mathematics chapter 10 - Sine and cosine formulae and their applications

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

Using RD Sharma Class 11 solutions Sine and cosine formulae and their applications exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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