Find the value of m 2 sin 1 2 π - n 2 sin 3 2 π + 2 m n sec π .

Let the given expression be E = m 2 sin 1 2 π - n 2 sin 3 2 π + 2 m n sec π . ⇒ E = m 2 sin π 2 - n 2 sin 3 π 2 + 2 m n sec π ⇒ E = m 2 1 - n 2 - 1 + 2 m n - 1 ⇒ E = m 2 + n 2 - 2 m n ⇒ E = ( m - n ) 2 ∴ The value of m 2 sin 1 2 π - n 2 sin 3 2 π + 2 m n sec π is ( m - n ) 2 .

In any triangle A B C , prove that : 2 b cos 2 C 2 + c cos 2 B 2 = a + b + c .

To prove: 2 b cos 2 C 2 + c cos 2 B 2 = a + b + c Given that, LHS = 2 b cos 2 C 2 + c cos 2 B 2 We know that, cos 2 θ = 1 + cos 2 θ 2 . ∴ 2 b cos 2 C 2 + c cos 2 B 2 = 2 b 1 + cos C 2 + c 1 + cos B 2 = b 1 + cos C + c 1 + cos B = b + c + ( b cos C + c cos B ) We know that, a = b cos C + c cos B . = a + b + c = RHS Hence, proved.

In any triangle A B C , prove that : sin A sin ( A + B ) = a c .

To prove: sin A sin ( A + B ) = a c Given that, A B C is a triangle i.e, A + B + C = 180 0 From, L.H.S, we have sin A sin ( A + B ) = sin A sin ( 180 0 - C ) = sin A sin C ∵ sin ( 180 - x ) = sin x We know that, a sin A = b sin B = c sin C = k ⇒ a k = sin A , b k = sin B , c k = sin C = a k c k = a c = R.H.S Hence, proved.

In any triangle A B C , prove that: a - b a + b = tan A - B 2 tan A + B 2 .

To prove: a - b a + b = tan A - B 2 tan A + B 2 Given that, A B C is a triangle i.e, A + B + C = 180 0 We know that, sine rule a sin A = b sin B = c sin C = k ⇒ a = k sin A , b = k sin B , c = k sin C . Therefore, L.H.S ⇒ a - b a + b = k sin A - k sin B k sin A + k sin B = sin A - sin B sin A + sin B We know that, sin C - sin D = 2 cos C + D 2 sin C - D 2 and sin C + sin D = 2 sin C + D 2 cos C - D 2 . = 2 cos A + B 2 sin A - B 2 2 sin A + B 2 cos A - B 2 = tan A - B 2 tan A + B 2 = R.H.S Hence, proved.

In any triangle A B C , prove that: b + c b - c = cot A 2 cot B - C 2 .

To prove: b + c b - c = cot A 2 cot B - C 2 and also given that, A B C is a triangle i.e, A + B + C = 180 0 → 1 We know that, a sin A = b sin B = c sin C = k ⇒ a = k sin A , b = k sin B , c = k sin C From L.H.S, we have b + c b - c = k sin B + k sin C k sin B - k sin C = sin B + sin C sin B - sin C We know that, sin C + sin D = 2 sin C + D 2 cos C - D 2 and sin C - sin D = 2 cos C + D 2 sin C - D 2 . = 2 sin B + C 2 cos B - C 2 2 cos B + C 2 sin B - C 2 = tan B + C 2 cot B - C 2 = tan 90 0 - A 2 cot B - C 2 (From 1 ) = cot A 2 cot B - C 2 = R.H.S Hence, proved.

In any triangle A B C , prove that: 1 + cos ( A - B ) cos C 1 + cos ( A - C ) cos B = a 2 + b 2 a 2 + c 2 .

To prove: 1 + cos ( A - B ) cos C 1 + cos ( A - C ) cos B = a 2 + b 2 a 2 + c 2 and also, given that, A B C is a triangle i.e, A + B + C = 180 0 → ( 1 ) From L.H.S, we have 1 + cos ( A - B ) cos C 1 + cos ( A - C ) cos B = 1 + cos ( A - B ) cos [ π - ( A + B ) ] 1 + cos ( A - C ) cos [ π - ( A + C ) ] = 1 - cos ( A - B ) cos ( A + B ) 1 - cos ( A - C ) cos ( A + C ) ∵ cos π - x = - cos x We know that, cos ( x + y ) cos ( x - y ) = cos 2 x - sin 2 y . = 1 - cos 2 A - sin 2 B 1 - cos 2 A - sin 2 C = 1 - cos 2 A + sin 2 B 1 - cos 2 A + sin 2 C = sin 2 A + sin 2 B sin 2 A + sin 2 C We know that, sine rule a sin A = b sin B = c sin C = k ⇒ a k = sin A , b k = sin B , c k = sin C = a 2 k 2 + b 2 k 2 a 2 k 2 + c 2 k 2 = a 2 + b 2 a 2 + c 2 = R.H.S Hence, proved.

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 3 - Trigonometry>CHAPTER TEST

M L Aggarwal Solutions for Chapter: Trigonometry, Exercise 11: CHAPTER TEST

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Trigonometry, Exercise 11: CHAPTER TEST

Attempt the practice questions on Chapter 3: Trigonometry, Exercise 11: CHAPTER TEST with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Trigonometry, Exercise 11: CHAPTER TEST with Hints & Solutions

EASY

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 3 - Trigonometry>CHAPTER TEST>Q 4

Find the value of $m^{2} \sin \frac{1}{2} π - n^{2} \sin \frac{3}{2} π + 2 m n \sec π .$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 3 - Trigonometry>CHAPTER TEST>Q 14

A railway carriage is travelling along a circular railway track of radius $1500$ metres with a speed of $66 km / hour$ . Find the angle in degrees turned by the engine in $10$ seconds.

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 3 - Trigonometry>CHAPTER TEST>Q 29

In any triangle $A B C$ , prove that : $2 (b \cos^{2} \frac{C}{2} + c \cos^{2} \frac{B}{2}) = a + b + c$ .