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

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 sin² A + sin² B = sin² C. show that the triangle is right-angled. 

In ∆ABC, if a², b² and c² 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 \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]

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 = 2b 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. 

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 sinA and sinB are the roots of the equation  \[c^2 x^2 - c\left( a + b \right)x + ab = 0\]  then find \[\angle C\]  

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. 

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. 

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 A, B, C. 

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)\ 

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

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

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

Mark the correct alternative in each of the following: 

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

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 

Mark the correct alternative in each of the following: 

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

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 

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 

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 

Mark the correct alternative in each of the following:

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