#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

## 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 Exercise 10.1 [Pages 12 - 14]

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

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

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

In triangle ABC, prove the following:

In triangle ABC, prove the following:

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

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In any triangle ABC, prove the following:

In triangle ABC, prove the following:

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

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

In triangle ABC, prove the following:

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

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

In triangle ABC, prove the following:

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

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

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

In ∆*ABC*, if sin^{2} *A* + sin^{2} *B* = sin^{2} *C*. show that the triangle is right-angled.

In ∆*A**BC*, if *a*^{2}, *b*^{2} and *c*^{2} are in A.P., prove that cot *A*, cot *B* and cot *C* are also in A.P.

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.

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.

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 \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]

If the sides *a*, *b* and *c* of ∆*ABC* are in H.P., prove that \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]

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

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

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.

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

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

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

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

In ∆ABC, prove the following:

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

In ∆ABC, prove the following*: *

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

In ∆ABC, prove the following*:*

*\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\] *

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

*a* cos *A *+* b* cos *B* + *c* cos *C* = 2*b *sin *A *sin *C *

In ∆ABC, prove the following:

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

In ∆ABC, prove the following:

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

In ∆ABC, prove the following:

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

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

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

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

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

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.

### RD Sharma solutions for Class 11 Mathematics Textbook Chapter 10 Sine and cosine formulae and their applications [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 \[\angle C = 60º\]

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*.

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*.

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 \[\sin A = \frac{3}{5}\]

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

In a ∆ABC, if sin*A* and sin*B* are the roots of the equation \[c^2 x^2 - c\left( a + b \right)x + ab = 0\] then find \[\angle C\]

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

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

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

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)\

In any ∆ABC, find the value of

\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]

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

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

Mark the correct alternative in each of the following:

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

\[\sqrt{3}\]

\[\sqrt{6}\]

\[\sqrt{9}\]

\[1 + \sqrt{2}\]

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

\[\frac{\pi}{6}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

\[\frac{2\pi}{3}\]

Mark the correct alternative in each of the following:

In any ∆ABC, 2(*bc* cos*A* + *ca *cos*B* + *ab* cos*C*) =

\[abc\]

\[a + b + c\]

\[a^2 + b^2 + c^2\]

\[\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\]

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

\[c^2 - 3c - 7 = 0\]

\[c^2 + 3c + 7 = 0\]

\[c^2 - 3c + 7 = 0\]

\[c^2 + 3c - 7 = 0\]

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

\[\frac{\pi}{3}\]

\[\frac{\pi}{6}\]

\[\frac{2\pi}{3}\]

\[\frac{\pi}{2}\]

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

\[a^2 + b^2 - c^2\]

\[c^2 + a^2 - b^2\]

\[b^2 - c^2 - a^2\]

\[c^2 - a^2 - b^2\]

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

## 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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